From pas at www.economia.unimi.it Wed Jan 4 09:55:34 2006 From: pas at www.economia.unimi.it (pas@www.economia.unimi.it) Date: Wed Jan 4 09:56:31 2006 Subject: [Pas] Probability Abstracts 90 Message-ID: January 4, 2006 Letter 90 Probability Abstract Service Abstracts from Nov-1-2005 to Dec-29-2005 --------------------------------------------------------------- 3796. PROPAGATION OF FLUCTUATIONS IN BIOCHEMICAL SYSTEMS, I: LINEAR SSC NETWORKS David Anderson and Jonathan Mattingly and H. Frederik Nijhout and Michael Reed We investigate the propagation of random fluctuations through biochemical networks in which the concentrations of species are large enough so that the unperturbed problem is well-described by ordinary differential equation. We characterize the behavior of variance as fluctuations propagate down chains, study the effect of side chains and feedback loops, and investigate the asymptotic behavior as one rate constant gets large. We also describe how the ideas can be applied to the study of methionine metabolism. http://front.math.ucdavis.edu/math.PR/0510642 --------------------------------------------------------------- 3797. TRANSPORTATION TO RANDOM ZEROES BY THE GRADIENT FLOW Fedor Nazarov and Mikhail Sodin and Alexander Volberg We show that the basins of zeroes under the gradient flow of the random potential U corresponding to a random Gaussian Entire Function f partition the complex plane into domains of equal area and that the probability that the diameter of a particular basin is greater than R is exponentially small in R. http://front.math.ucdavis.edu/math.CV/0510654 --------------------------------------------------------------- 3798. NO MULTIPLE COLLISIONS FOR MUTUALLY REPELLING BROWNIAN PARTICLES Emmanuel C\'{e}pa (MAPMO) and Dominique L\'{e}pingle (MAPMO) Brownian particles in electrostatic interaction may pairwise collide when the interaction parameter is small. But multiple collisions are never possible. http://front.math.ucdavis.edu/math.PR/0511445 --------------------------------------------------------------- 3799. A PERMUTATION TEST FOR MATCHING AND ITS ASYMPTOTIC DISTRIBUTION Larry Goldstein and Yosef Rinott We consider a permutation method for testing whether observations given in their natural pairing exhibit an unusual level of similarity in situations where any two observations may be similar at some unknown baseline level. Under a null hypotheses where there is no distinguished pairing of the observations, a normal approximation with explicit bounds and rates is presented for determining approximate critical test levels. http://front.math.ucdavis.edu/math.ST/0511427 --------------------------------------------------------------- 3800. COMBINATORICS AND DISTRIBUTIONS OF PARTIAL INJECTIONS Olexandr Ganyushkin and Volodymyr Mazorchuk We obtain several combinatorial results about chains, cycles and orbits of the elements of the symmetric inverse semigroup $\IS_n$ and the set $T_n$ of nilpotent elements in $\IS_n$. We also get some estimates for the growth of $|\IS_n|$ and $|T_n|$, and study random products of elements from $ \IS_n$. http://front.math.ucdavis.edu/math.CO/0511431 --------------------------------------------------------------- 3801. MULTIPLE ORTHOGONAL POLYNOMIALS OF MIXED TYPE AND NON- INTERSECTING BROWNIAN MOTIONS E. Daems and A.B.J. Kuijlaars We present a generalization of multiple orthogonal polynomials of type I and type II, which we call multiple orthogonal polynomials of mixed type. Some basic properties are formulated, and a Riemann-Hilbert problem for the multiple orthogonal polynomials of mixed type is given. We derive a Christoffel-Darboux formula for these polynomials using the solution of the Riemann-Hilbert problem. The main motivation for studying these polynomials comes from a model of non-intersecting one-dimensional Brownian motions with a given number of starting points and endpoints. The correlation kernel for the positions of the Brownian paths at any intermediate time coincides with the Christoffel-Darboux kernel for the multiple orthogonal polynomials of mixed type with respect to Gaussian weights. http://front.math.ucdavis.edu/math.CA/0511470 --------------------------------------------------------------- 3802. AN ORIENTED COMPETITION MODEL ON Z_{+}^2 George Kordzakhia and Steven Lalley We consider a two-type oriented competition model on the first quadrant of the two-dimensional integer lattice. Each vertex of the space may contain only one particle of either Red type or Blue type. A vertex flips to the color of a randomly chosen southwest nearest neighbor at exponential rate 2. At time zero there is one Red particle located at (1,0) and one Blue particle located at (0,1). The main result is a partial shape theorem: Denote by R(t) and B(t) the red and blue regions at time t. Then (i) eventually the upper half of the unit square contains no points of B(t)=t, and the lower half no points of R (t)=t; and (ii) with positive probability there are angular sectors rooted at (1,1) that are eventually either red or blue. The second result is contingent on the uniform curvature of the boundary of the corresponding Richardson shape. http://front.math.ucdavis.edu/math.PR/0511504 --------------------------------------------------------------- 3803. BERRY ESSEEN BOUNDS FOR COMBINATORIAL CENTRAL LIMIT THEOREMS AND PATTERN OCCURRENCES, USING ZERO AND SIZE BIASING Larry Goldstein Berry Esseen type bounds to the normal, based on zero- and size-bias couplings, are derived using Stein's method. The zero biasing bounds are illustrated with an application to combinatorial central limit theorems where the random permutation has either the uniform distribution or one which is constant over permutations with the same cycle type and having no fixed points. The size biasing bounds are applied to the occurrences of fixed relatively ordered sub-sequences (such as rising sequences) in a random permutation, and to the occurrences of patterns, extreme values, and subgraphs on finite graphs. http://front.math.ucdavis.edu/math.PR/0511510 --------------------------------------------------------------- 3804. RANDOM DENSE COUNTABLE SETS: CHARACTERIZATION BY INDEPENDENCE Boris Tsirelson A random dense countable set is characterized (in distribution) by independence and stationarity. Two examples are `Brownian local minima' and `unordered infinite sample'. They are identically distributed; the former ad hoc proof of this fact is now superseded by a general result. http://front.math.ucdavis.edu/math.PR/0511011 --------------------------------------------------------------- 3805. STOCHASTIC INTEGRAL WITH RESPECT TO CYLINDRICAL WIENER PROCESS Anna Karczewska This paper is devoted to a construction of the stochastic It\^o integral with respect to infinite dimensional cylindrical Wiener process. The construction given is an alternative one to that introduced by DaPrato and Zabczyk [3]. The connection of the introduced integral with the integral defined by Walsh [9] is provided as well. http://front.math.ucdavis.edu/math.PR/0511512 --------------------------------------------------------------- 3806. RANDOM TREES AND APPLICATIONS Jean-Francois Le Gall We discuss several connections between discrete and continuous random trees. In the discrete setting, we focus on Galton-Watson trees under various conditionings. In particular, we present a simple approach to Aldous' theorem giving the convergence in distribution of the contour process of conditioned Galton-Watson trees towards the normalized Brownian excursion. We also briefly discuss applications to combinatorial trees. In the continuous setting, we use the formalism of real trees, which yields an elegant formulation of the convergence of rescaled discrete trees towards continuous objects. We explain the coding of real trees by functions, which is a continuous version of the well-known coding of discrete trees by Dyck paths. We pay special attention to random real trees coded by Brownian excursions, and in a particular we provide a simple derivation of the marginal distributions of the CRT. The last section is an introduction to the theory of the Brownian snake, which combines the genealogical structure of random real trees with independent spatial motions. We introduce exit measures for the Brownian snake and we present some applications to a class of semilinear partial differential equations. http://front.math.ucdavis.edu/math.PR/0511515 --------------------------------------------------------------- 3807. EXPONENTIAL FUNCTIONALS OF BROWNIAN MOTION, I: PROBABILITY LAWS AT FIXED TIME Hiroyuki Matsumoto Marc Yor This paper is the first part of our survey on various results about the distribution of exponential type Brownian functionals defined as an integral over time of geometric Brownian motion. Several related topics are also mentioned. http://front.math.ucdavis.edu/math.PR/0511517 --------------------------------------------------------------- 3808. EXPONENTIAL FUNCTIONALS OF BROWNIAN MOTION, II: SOME RELATED DIFFUSION PROCESSES Hiroyuki Matsumoto Marc Yor This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of L\'evy's and Pitman's theorems are discussed. http://front.math.ucdavis.edu/math.PR/0511519 --------------------------------------------------------------- 3809. A VARIATION EMBEDDING THEOREM AND APPLICATIONS Peter Friz and Nicolas Victoir Fractional Sobolev spaces, also known as Besov or Slobodetzki spaces, arise in many areas of analysis, stochastic analysis in particular. We prove an embedding into certain q-variation spaces and discuss a few applications. First we show q-variation regularity of Cameron-Martin paths associated to fractional Brownian motion and other Volterra processes. This is useful, for instance, to establish large deviations for enhanced fractional Brownian motion. Second, the q-variation embedding, combined with results of rough path theory, provides a different route to a regularity result for stochastic differential equations by Kusuoka. Third, the embedding theorem works in a non-commutative setting and can be used to establish Hoelder/variation regularity of rough paths. http://front.math.ucdavis.edu/math.PR/0511520 --------------------------------------------------------------- 3810. GIANT COMPONENTS IN BIASED GRAPH PROCESSES Gideon Amir and Ori Gurel-Gurevich and Eyal Lubetzky and Amit Singer A random graph process, $\Gorg[1](n)$, is a sequence of graphs on $n$ vertices which begins with the edgeless graph, and where at each step a single edge is added according to a uniform distribution on the missing edges. It is well known that in such a process a giant component (of linear size) typically emerges after $(1+o(1))\frac{n}{2}$ edges (a phenomenon known as ``the double jump''), i.e., at time $t=1$ when using a timescale of $n/2$ edges in each step. We consider a generalization of this process, $\Gorg[K](n)$, which gives a weight of size 1 to missing edges between pairs of isolated vertices, and a weight of size $K \in [0,\infty)$ otherwise. This corresponds to a case where links are added between $n$ initially isolated settlements, where the probability of a new link in each step is biased according to whether or not its two endpoint settlements are still isolated. Combining methods of \cite{SpencerWormald} with analytical techniques, we describe the typical emerging time of a giant component in this process, $t_c(K)$, as the singularity point of a solution to a set of differential equations. We proceed to analyze these differential equations and obtain properties of $\Gorg$, and in particular, we show that $t_c(K)$ strictly decreases from 3/2 to 0 as $K$ increases from 0 to $\infty$, and that $t_c(K) = \frac{4}{\sqrt{3K}}(1 + o(1))$. Numerical approximations of the differential equations agree both with computer simulations of the process $\Gorg (n)$ and with the analytical results. http://front.math.ucdavis.edu/math.PR/0511526 --------------------------------------------------------------- 3811. FOURIER TRANSFORM OF A GAUSSIAN MEASURE ON THE HEISENBERG GROUP Matyas Barczy and Gyula Pap An explicit formula is derived for the Fourier transform of a Gaussian measure on the Heisenberg group at the Schrodinger representation. Using this explicit formula, necessary and sufficient conditions are given for the convolution of two Gaussian measures to be a Gaussian measure. http://front.math.ucdavis.edu/math.PR/0511016 --------------------------------------------------------------- 3812. THE SPATIAL $\LAMBDA$-COALESCENT Vlada Limic and Anja Sturm This paper extends the notion of the $\la$-coalescent of Pitman (1999) to the spatial setting. The partition elements of the spatial $\Lambda$- coalescent migrate in a (finite) geographical space and may only coalesce if located at the same site of the space. We characterize the $\Lambda$-coalescents that come down from infinity, in an analogous way to Schweinsberg (2000). Surprisingly, all spatial coalescents that come down from infinity, also come down from infinity in a uniform way. This enables us to study space-time asymptotics of spatial $\Lambda$-coalescents on large tori in $d\ge 3$ dimensions. Our results generalize and strengthen those of Greven et al. (2005), who studied the spatial Kingman coalescent in this context. http://front.math.ucdavis.edu/math.PR/0511536 --------------------------------------------------------------- 3813. THE REALIZATION OF POSITIVE RANDOM VARIABLES VIA ABSOLUTELY CONTINUOUS TRANSFORMATIONS OF MEASURE ON WIENER SPACE D. Feyel and A.S. Ustunel and M. Zakai Let \mu be a Gaussian measure on some measurable space {W = {w}, \calB (W)} and let \nu be a measure on the same space which is absolutely continuous with respect to \nu. The paper surveys results on the problem of constructing a transformation T on the W space such that Tw = w+u(w) where u takes values in the Cameron-Martin space and the image of \mu under T is \mu. In addition we ask for the existence of transformations T belonging to some particular classes. http://front.math.ucdavis.edu/math.PR/0511545 --------------------------------------------------------------- 3814. RANDOM WALK MODELS AND PROBABILISTIC TECHNIQUES FOR INHOMOGENEOUS POLYMER CHAINS Francesco Caravenna Modeling of polymer chains has received a lot of attention in mathematics. In fact, probabilistic models that naturally arise in statistical mechanics have been widely studied by mathematicians for the very challenging and novel problems that they pose. The physical situation that we consider in this thesis is that of a polymer in the proximity of an interface between two selective solvents, in the case when the interaction of the monomers with the solvents and the interface may vary from monomer to monomer (inhomogeneous polymer). In interesting cases thee is a phase transition between a state in which the polymer sticks very close to the interface (localized regime) and a state in which it wanders away from it (delocalized regime). The mechanism underlying such a transition is an energy/entropy competition. Our task has been to study random walk models of polymer chains with the purpose of understanding this competition in a deep and quantitative way. Despite the fact that the definition of these models is extremely elementary, their analysis is not simple at all, and several interesting questions are still open. In this Ph.D. thesis we present new results that answer some of these questions. The analysis performed has required the application of a wide range of techniques, including large deviations, concentration inequalities, renewal theory, fluctuation theory for random walks. A numerical and statistical study has been performed too. Finally we prove a local limit theorem for random walks conditioned to stay positive. http://front.math.ucdavis.edu/math.PR/0511561 --------------------------------------------------------------- 3815. ON CONSTRAINED ANNEALED BOUNDS FOR PINNING AND WETTING MODELS Francesco Caravenna and Giambattista Giacomin The free energy of quenched disordered systems is bounded above by the free energy of the corresponding annealed system. This bound may be improved by applying the annealing procedure, which is just Jensen inequality, after having modified the Hamiltonian in a way that the quenched expressions are left unchanged. This procedure is often viewed as a partial annealing or as a constrained annealing, in the sense that the term that is added may be interpreted as a Lagrange multiplier on the disorder variables. In this note we point out that, for a family of models, some of which have attracted much attention, the multipliers of the form of empirical averages of local functions cannot improve on the basic annealed bound from the viewpoint of characterizing the phase diagram. This class of multipliers is the one that is suitable for computations and it is often believed that in this class one can approximate arbitrarily well the quenched free energy. http://front.math.ucdavis.edu/math.PR/0511562 --------------------------------------------------------------- 3816. A MODIFIED VERSION OF FROZEN PERCOLATION ON THE BINARY TREE R.Brouwer We consider the following, intuitively described process: at time zero, all sites of a binary tree are at rest. Each site becomes activated at a random uniform [0,1] time, independent of the other sites. As soon as a site is in an infinite cluster of activated sites, this cluster of activated sites freezes. The main question is whether a process like this exists. Aldous [Ald00] proved that this is the case for a slightly different version of frozen percolation. In this paper we construct a process that fits the intuitive description and discuss some properties. http://front.math.ucdavis.edu/math.PR/0511021 --------------------------------------------------------------- 3817. DIRECTED PERCOLATION IN TWO DIMENSIONS: AN EXACT SOLUTION L. C. Chen and F. Y. Wu We consider a directed percolation process on an ${\cal M}$ x ${\cal N}$ rectangular lattice whose vertical edges are directed upward with an occupation probability y and horizontal edges directed toward the right with occupation probabilities x and 1 in alternate rows. We deduce a closed-form expression for the percolation probability P(x,y), the probability that one or more directed paths connect the lower-left and upper-right corner sites of the lattice. It is shown that P(x,y) is critical in the aspect ratio $a = {\cal M}/{\cal N}$ at a value $a_c =[1-y^2-x(1-y)^2]/2y^2$ where P(x,y) is discontinuous, and the critical exponent of the correlation length for $a < a_c$ is $\nu=2$. http://front.math.ucdavis.edu/cond-mat/0511296 --------------------------------------------------------------- 3818. ON THE LIMITING DISTRIBUTION FOR THE LONGEST ALTERNATING SEQUENCE IN A RANDOM PERMUTATION Harold Widom Recently Richard Stanley initiated a study of the distribution of the length as(w) of the longest alternating subsequence in a random permutation w from the symmetric group $S_n$. Among other things he found an explicit formula for the generating function (on n and k) for the probability that as(w) is at most k and conjectured that the distribution, suitably centered and normalized, tended to a Gaussian with variance 8/45. In this note we present a proof of the conjecture based on the generating function. http://front.math.ucdavis.edu/math.CO/0511533 --------------------------------------------------------------- 3819. LINEAR FUNCTIONS ON THE CLASSICAL MATRIX GROUPS Elizabeth Meckes Let $M$ be a random matrix in the orthogonal group $\O_n$, distributed according to Haar measure, and let $A$ be a fixed $n\times n$ matrix over $\R$ such that $\tr(AA^t)=n$. Then the total variation distance of the random variable $\tr(AM)$ to standard normal is bounded by $2\sqrt{3}/(n-1) $, and this rate is sharp up to the constant. Analogous results are obtained for $M$ a random unitary matrix and $A$ a fixed $n\times n$ matrix over $\C$. The proofs are via an improvement of Stein's method of exchangeable pairs which makes use of the continuous nature of the symmetries of the classical matrix groups. http://front.math.ucdavis.edu/math.PR/0509441 --------------------------------------------------------------- 3820. ZERO BIASING AND A DISCRETE CENTRAL LIMIT THEOREM Larry Goldstein and Aihua Xia We introduce a new family of distributions to approximate $\prob(W\in A)$ for $A\subset\{...,-2,-1,0,1,2,...\}$ and $W$ a sum of independent integer-valued random variables $\xi_1$, $\xi_2$, $...$, $\xi_n$ with finite second moments, where with large probability $W$ is not concentrated on a lattice of span greater than 1. The well-known Berry--Esseen theorem states that for $Z$ a normal random variable with mean $\mean(W)$ and variance $\var(W)$, $ \prob(Z \in A)$ provides a good approximation to $\prob(W \in A)$ for $A$ of the form $(-\infty,x]$. However, for more general $A$ such as the set of all even numbers, the normal approximation becomes unsatisfactory and it is desirable to have an appropriate discrete, non-normal, distribution which approximates $W$ in total variation, and a discrete version of the Berry--Esseen theorem to bound the error. In this paper, using the concept of zero biasing for discrete random variables [cf Goldstein and Reinert (2005)], we introduce a new family of discrete distributions and provide a discrete version of the Berry--Esseen theorem showing how members of the family approximate the distribution of a sum $W$ of integer valued variables in total variation. http://front.math.ucdavis.edu/math.PR/0509444 --------------------------------------------------------------- 3821. ON A CLASS OF STOCHASTIC SEMILINEAR PDE'S Luigi Manca We consider stochastic semilinear partial differential equations with Lipschitz nonlinear terms. We prove existence and uniqueness of an invariant measure and the existence of a solution for the corresponding Kolmogorov equation in the space $L^2(H;\nu)$, where $\nu$ is the invariant measure. We also prove the closability of the derivative operator and an integration by parts formula. Finally, under boundness conditions on the nonlinear term, we prove a Poincar\'e inequality, a logarithmic Sobolev inequality and the ipercontractivity of the transition semigroup. http://front.math.ucdavis.edu/math.PR/0509446 --------------------------------------------------------------- 3822. A CENTRAL LIMIT THEOREM AND HIGHER ORDER RESULTS FOR THE ANGULAR BISPECTRUM D. Marinucci The angular bispectrum of spherical random fields has recently gained an enormous importance, especially in connection with statistical inference on cosmological data. In this paper, we provide expressions for its moments of arbitrary order and we use these results to establish a multivariate central limit theorem and higher order approximations. The results rely upon combinatorial methods from graph theory and a detailed investigation for the asymptotic behaviour of Clebsch-Gordan coefficients; the latter are widely used in representation theory and quantum theory of angular momentum. http://front.math.ucdavis.edu/math.PR/0509430 --------------------------------------------------------------- 3823. FLUCTUATIONS OF THE FRONT IN A STOCHASTIC COMBUSTION MODEL Francis Comets and Jeremy Quastel and Alejandro F. Ramirez We consider an interacting particle system on the one dimensional lattice $\bf Z$ modeling combustion. The process depends on two integer parameters $2\le ad/(d+\alpha), which coincides with the case of finite variance branching (\beta=1), and another one for \beta\leq d/(d+\alpha), where the long range dependence depends on the value of \beta. The long range dependence is characterized by a dependence exponent \kappa which describes the asymptotic behavior of the codifference of increments of \xi on intervals far apart, and which is d/\alpha for the first case and (1+\beta-d/(d+\alpha))d/\alpha for the second one. The convergence proofs use techniques of S'(R^d)-valued processes. http://front.math.ucdavis.edu/math.PR/0511739 --------------------------------------------------------------- 3833. THE PROCESS OF MOST RECENT COMMON ANCESTORS IN AN EVOLVING COALESCENT P. Pfaffelhuber and A. Wakolbinger In a population of constant size, whose family sizes evolve as Wright- Fisher diffusions, all individuals alive at time $t$ have a most recent common ancestor (MRCA) who lived at time $A(t)$, say. The process $(A(t))$ has piecewise constant paths. At each jump time $E_n$, a new MRCA takes over, who lived at time $B_n:=A(E_n)$. We construct the random sequence $(B_n, E_n)$ in terms of a look-down process and investigate its dynamics as well as that of $(A(t))$. In particular, we find the joint distribution of the waiting time from $t$ to the next MRCA change and of the time when this next MRCA will have lived. http://front.math.ucdavis.edu/math.PR/0511743 --------------------------------------------------------------- 3834. THE FULL BROWNIAN WEB AS SCALING LIMIT OF STOCHASTIC FLOWS Luiz Renato Fontes Charles M. Newman In this paper we construct an object which we call the full Brownian web (FBW) and prove that the collection of all space-time trajectories of a class of one-dimensional stochastic flows converges weakly, under diffusive rescaling, to the FBW. The (forward) paths of the FBW include the coalescing Brownian motions of the ordinary Brownian web along with bifurcating paths. Convergence of rescaled stochastic flows to the FBW follows from general characterization and convergence theorems that we present here combined with earlier results of Piterbarg. http://front.math.ucdavis.edu/math.PR/0511029 --------------------------------------------------------------- 3835. OCCUPATION TIME FLUCTUATIONS OF AN INFINITE VARIANCE BRANCHING SYSTEM IN LARGE DIMENSIONS Tomasz Bojdecki and Luis G. Gorostiza and Anna Talarczyk We prove limit theorems for rescaled occupation time fluctuations of a (d,alpha,beta)-branching particle system (particles moving in R^d according to a spherically symmetric alpha-stable Levy process, (1+beta)-branching, 0alpha(1+beta)/beta. The fluctuation processes are continuous but their limits are stable processes with independent increments, which have jumps. The convergence is in the sense of finite-dimensional distributions, and also of space-time random fields (tightness does not hold in the usual Skorohod topology). The results are in sharp contrast with those for intermediate dimensions, alpha/beta < d < d(1+beta)/beta, where the limit process is continuous and has long range dependence (this case is studied by Bojdecki et al, 2005). The limit process is measure-valued for the critical dimension, and S'(R^d)-valued for large dimensions. We also raise some questions of interpretation of the different types of dimension-dependent results obtained in the present and previous papers in terms of properties of the particle system. http://front.math.ucdavis.edu/math.PR/0511745 --------------------------------------------------------------- 3836. ASYMPTOTIC BEHAVIOR OF EDGE-REINFORCED RANDOM WALKS Franz Merkl and Silke Rolles In this article, we study linearly edge-reinforced random walk on general multi-level ladders for large initial edge weights. For infinite ladders, we show that the process can be represented as a random walk in a random environment, given by random weights on the edges. The edge weights decay exponentially in space. The process converges to a stationary process. We provide asymptotic bounds for the range of the random walker up to a given time, showing that it localizes much more than an ordinary random walker. The random environment is described in terms of an infinite-volume Gibbs measure. http://front.math.ucdavis.edu/math.PR/0511750 --------------------------------------------------------------- 3837. QUANTITATIVE CONCENTRATION INEQUALITIES ON SAMPLE PATH SPACE FOR MEAN FIELD INTERACTION Fran\c{c}ois Bolley (UMPA-ENSL) We consider a system of particles experiencing diffusion and mean field interaction, and study its behaviour when the number of particles goes to infinity. We derive non-asymptotic large deviation bounds measuring the concentration of the empirical measure of the paths of the particles around its limit. The method is based on a coupling argument, strong integrability estimates on the paths in Holder norm, and some general concentration result for the empirical measure of identically distributed independent paths. http://front.math.ucdavis.edu/math.PR/0511752 --------------------------------------------------------------- 3838. ROSENTHAL TYPE INEQUALITIES FOR FREE CHAOS Marius Junge and Javier Parcet and Quanhua Xu Let $\mathcal{A}$ denote the reduced amalgamated free product of a family $\mathsf{A}_1, \mathsf{A}_2, ..., \mathsf{A}_n$ of von Neumann algebras over a von Neumann subalgebra $\Be$ with respect to normal faithful conditional expectations $\Es_k: \mathsf{A}_k \to \Be$. We investigate the norm in $L_p(\Al)$ of homogeneous polynomials of a given degree $d$. We first generalize Voiculescu's inequality to arbitrary degree $d \ge 1$ and indices $1 \le p \le \infty$. This can be regarded as a free analogue of the classical Rosenthal inequality. Our second result is a length-reduction formula from which we generalize recent results of Pisier, Ricard and the authors. All constants in our estimates are independent of $n$ so that we may consider infinitely many free factors. As applications, we study square functions of free martingales. More precisely we show that, in contrast with the Khintchine and Rosenthal inequalities, the free analogue of the Burkholder-Gundy inequalities does not hold on $L_\infty(\Al)$. At the end of the paper we also consider Khintchine type inequalities for Shlyakhtenko's generalized circular systems. http://front.math.ucdavis.edu/math.OA/0511732 --------------------------------------------------------------- 3839. SPATIAL AND NON-SPATIAL STOCHASTIC MODELS FOR IMMUNE RESPONSE Rinaldo Schinazi and Jason Schweinsberg We study some simple mathematical models designed to test the following hypothesis: can a pathogen escape the immune system only because of its high probability of mutation? We propose both spatial and non-spatial models. In all of our models, we assume that pathogens can mutate, leading to the appearance of new types of pathogens. We also assume that the immune system is able to get rid of all the pathogens of a given type at once but that it recognizes only one type at a time. http://front.math.ucdavis.edu/math.PR/0512009 --------------------------------------------------------------- 3840. COLOURING POWERS OF CYCLES FROM RANDOM LISTS Michael Krivelevich and Asaf Nachmias Let $C_n^k$ be the $k$-th power of a cycle on $n$ vertices (i.e. the vertices of $C_n^k$ are those of the $n$-cycle, and two vertices are connected by an edge if their distance along the cycle is at most $k$). For each vertex draw uniformly at random a subset of size $c$ from a base set $S$ of size $s=s(n)$. In this paper we solve the problem of determining the asymptotic probability of the existence of a proper colouring from the lists for all fixed values of $c,k$, and growing $n$. http://front.math.ucdavis.edu/math.CO/0512004 --------------------------------------------------------------- 3841. COLOURING COMPLETE BIPARTITE GRAPHS FROM RANDOM LISTS Michael Krivelevich and Asaf Nachmias Let $K_{n,n}$ be the complete bipartite graph with $n$ vertices in each side. For each vertex draw uniformly at random a list of size $k$ from a base set $S$ of size $s=s(n)$. In this paper we estimate the asymptotic probability of the existence of a proper colouring from the random lists for all fixed values of $k$ and growing $n$. We show that this property exhibits a sharp threshold for $k\geq 2$ and the location of the threshold is precisely $s(n)=2n$ for $k=2$, and approximately $s(n)=\frac{n}{2^{k-1}\ln 2}$ for $k\geq 3$. http://front.math.ucdavis.edu/math.CO/0512010 --------------------------------------------------------------- 3842. INCREASING AND DECREASING SUBSEQUENCES OF PERMUTATIONS AND THEIR VARIANTS Richard P. Stanley We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1,2,...,n was obtained by Vershik-Kerov and (almost) by Logan-Shepp. The entire limiting distribution of is(w) was then determined by Baik, Deift, and Johansson. These techniques can be applied to other classes of permutations, such as involutions, and are related to the distribution of eigenvalues of elements of the classical groups. A number of generalizations and variations of increasing/decreasing subsequences are discussed, including the theory of pattern avoidance, unimodal and alternating subsequences, and crossings and nestings of matchings and set partitions. http://front.math.ucdavis.edu/math.CO/0512035 --------------------------------------------------------------- 3843. LIMIT VELOCITY AND ZERO-ONE LAWS FOR DIFFUSIONS IN RANDOM ENVIRONMENT Laurent Goergen This article is accepted for publication in the "Annals of Applied Probability". We prove that multi-dimensional diffusions in random environment have a limiting velocity which takes at most two different values. Further, in the two-dimensional case we show that for any direction, the probability to escape to infinity in this direction equals either zero or one. Combined with our results on the limiting velocity, this implies a strong law of large numbers in two dimensions. http://front.math.ucdavis.edu/math.PR/0512061 --------------------------------------------------------------- 3844. A MICROSCOPIC INTERPRETATION FOR ADAPTIVE DYNAMICS TRAIT SUBSTITUTION SEQUENCE MODELS Nicolas Champagnat (WIAS) We consider an interacting particle Markov process for Darwinian evolution in an asexual population with non-constant population size, involving a linear birth rate, a density-dependent logistic death rate, and a probability $\mu$ of mutation at each birth event. We introduce a renormalization parameter $K$ scaling the size of the population, which leads, when $K\to+\infty$, to a deterministic dynamics for the density of individuals holding a given trait. By combining in a non-standard way the limits of large population ($K\to+ \infty$) and of small mutations ($\mu\to 0$), we prove that a time scales separation between the birth and death events and the mutation events occurs and that the interacting particle microscopic process converges for finite dimensional distributions to the biological model of evolution known as the ``monomorphic trait substitution sequence'' model of adaptive dynamics, which describes the Darwinian evolution in an asexual population as a Markov jump process in the trait space. http://front.math.ucdavis.edu/math.PR/0512063 --------------------------------------------------------------- 3845. FUNCTIONAL INEQUALITIES FOR PARTICLE SYSTEMS ON POLISH SPACES Michael R\"ockner and Feng-Yu Wang Various Poincare-Sobolev type inequalities are studied for a reaction-diffusion model of particle systems on Polish spaces. The systems we consider consist of finite particles which are killed or produced at certain rates, while particles in the system move on the Polish space interacting with one another (i.e. diffusion). Thus, the corresponding Dirichlet form, which we call reaction-diffusion Dirichlet form, consists of two parts: the diffusion part induced by certain Markov processes on the product spaces $E^n (n \geq 1)$ which determine the motion of particles, and the reaction part induced by a $Q$-process on $\mathbb Z_+$ and a sequence of reference probability measures, where the $Q$-process determines the variation of the number of particles and the reference measures describe the locations of newly produced particles. We prove that the validity of Poincare and weak Poincare inequalities are essentially due to the pure reaction part, i.e. either of these inequalities holds if and only if it holds for the pure reaction Dirichlet form, or equivalently, for the corresponding $Q$-process. But under a mild condition, stronger inequalities rely on both parts: the reaction-diffusion Dirichlet form satisfies a super Poincare inequality (e.g. the log-Sobolev inequality) if and only if so do both the corresponding $Q$-process and the diffusion part. Explicit estimates of constants in the inequalities are derived. Finally, some specific examples are presented to illustrate the main results. http://front.math.ucdavis.edu/math.PR/0512100 --------------------------------------------------------------- 3846. JOINT ASYMPTOTIC BEHAVIOR OF LOCAL AND OCCUPATION TIMES Endre Cs\'{a}ki and Ant\'{o}nia F\"{o}ldes and P\'al R\'ev\'esz Considering a simple symmetric random walk in dimension $d\geq 3$, we study the almost sure joint asymptotic behavior of two objects: first the local times of a pair of neighboring points, then the local time of a point and the occupation time of the surface of the unit ball around it. http://front.math.ucdavis.edu/math.PR/0511049 --------------------------------------------------------------- 3847. INFINITELY DIVISIBLE DISTRIBUTIONS FOR RECTANGULAR FREE CONVOLUTION: CLASSIFICATION AND MATRICIAL INTERPRETATION Florent Benaych-Georges (DMA) In a previous paper (called "Rectangular random matrices. Related covolution"), we defined, for $\lambda \in [0,1]$, the rectangular free convolution with ratio $\lambda$. Here, we investigate the related notion of infinite divisiblity, which happens to be closely related the classical infinite divisibility: there exists a bijection between the set of classical symmetric infinitely divisible distributions and the set of infinitely divisible distributions with respect to this convolution, which preserves limit theorems. We give an interpretation of this correspondance in term of random matrices: we construct distributions on sets of complex rectangular matrices which give rise to random matrices with singular laws (i.e. uniform distributions on their singular values) going from the symmetric classical infinitely divisible distributions to their images by the previously mentioned bijection when the dimensions go from one to infinity in a ratio $ \lambda$. http://front.math.ucdavis.edu/math.OA/0512080 --------------------------------------------------------------- 3848. RECTANGULAR RANDOM MATRICES, RELATED FREE ENTROPY AND FREE FISHER'S INFORMATION Florent Benaych-Georges (DMA) We prove that independent rectangular random matrices, when embedded in a space of larger square matrices, are asymptotically free with amalgamation over a commutative finite dimensional subalgebra $D$ (under an hypothesis of unitary invariance). Then we consider elements of a finite von Neumann algebra containing $D$, which have kernel and range projection in $D$. We associate them a free entropy with the microstates approach, and a free Fisher's information with the conjugate variables approach. Both give rise to optimization problems whose solutions involve freeness with amalgamation over $D$. It could be a first proposition for the study of operators between different Hilbert spaces with the tools of free probability. As an application, we prove a result of freeness with amalgamation between the two parts of the polar decomposition of $R$-diagonal elements with non trivial kernel. http://front.math.ucdavis.edu/math.OA/0512081 --------------------------------------------------------------- 3849. OPTIMAL CONTROL OF A LARGE DAM Vyacheslav M. Abramov A large dam model is an object of study of this paper. The parameters $L^{lower}$ and $L^{upper}$ are its lower and upper levels, $L=L^{upper}-L^{lower}$ is large, and if a current level of water is between these bounds, then the dam is assumed to be in normal state. Passage one or other bound leads to damage. It is assumed that input stream of water is described by a Poisson process, while the output stream is state- dependent (the exact formulation of the problem is given in the paper). Let $L_t$ denote the dam level at time $t$, and let $p_1=\lim_{t\to\infty}\mathbf{P}\{L_t= L^{lower}\}$, $p_2=\lim_{t\to\infty}\mathbf{P}\{L_t> L^{upper}\}$ exist. Then the expected long-run damage $J=p_1J_1+p_2J_2$ for the long time interval $T$ proportional to $L$ ($J_1$ and $J_2$ are the corresponding damage costs per time $T$ associated with passage the bounds) is a performance measure, and the aim of the paper is to choose the parameter of output stream (exactly specified in the paper) minimizing $J$. http://front.math.ucdavis.edu/math.PR/0512118 --------------------------------------------------------------- 3850. QUASI-PRODUCT FORMS FOR LEVY-DRIVEN FLUID NETWORKS K. Debicki and A. B. Dieker and T. Rolski We study stochastic tree fluid networks driven by a multidimensional Levy process. We are interested in (the joint distribution of) the steady- state content in each of the buffers, the busy periods, and the idle periods. To investigate these fluid networks, we relate the above three quantities to fluctuations of the input Levy process by solving a multidimensional Skorokhod problem. This leads to the analysis of the distribution of the componentwise maximums, the corresponding epochs at which they are attained, and the beginning of the first last-passage excursion. Using the notion of splitting times, we are able to find their Laplace transforms. It turns out that, if the components of the Levy process are `ordered', the Laplace transform has a so-called quasi-product form. The theory is illustrated by working out special cases, such as tandem networks and priority queues. http://front.math.ucdavis.edu/math.PR/0512119 --------------------------------------------------------------- 3851. ASYMPYOTIC EXPANSIONS FOR INFINITE WEIGHTED CONVOLUTIONS OF LIGHT SUBEXPONENTIAL DISTRIBUTIONS Ph. Barbe (CNRS) and W.P. McCormick (UGA) We establish some asymptotic expansions for infinite weighted convolutions of distributions having light subexponential tails. Examples are presented, some showing that in order to obtain an expansion with two significant terms, one needs to have a general way to calculate higher order expansions, due to possible cancellations of terms. An algebraic methodology is employed to obtain the results. http://front.math.ucdavis.edu/math.PR/0512141 --------------------------------------------------------------- 3852. BACKWARD STOCHATIC DIFFERENTIAL EQUATIONS II Fabrice Blache (LMA-Clermont) In a preceding article, we have studied a generalization of the problem of finding a martingale on a manifold whose terminal value is known. This article completes the results obtained in the first article by providing uniqueness and existence theorems in a general framework (in particular if positive curvatures are allowed), still using differential geometry tools. http://front.math.ucdavis.edu/math.PR/0512145 --------------------------------------------------------------- 3853. DISTRIBUTION OF EIGENVALUES FOR THE ENSEMBLE OF REAL SYMMETRIC PALINDROMIC TOEPLITZ MATRICES Adam Massey and Steven J. Miller and John Sinsheimer Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the density of normalized eigenvalues) converges (weakly and almost surely), independent of p, to a distribution which is almost the Gaussian. The deviations from Gaussian behavior can be interpreted as arising from obstructions to solutions of Diophantine equations. We show that these obstructions vanish if instead one considers real symmetric palindromic Toeplitz matrices (matrices where the first row is a palindrome), and the resulting spectral measures converge (weakly and almost surely) to the Gaussian. http://front.math.ucdavis.edu/math.PR/0512146 --------------------------------------------------------------- 3854. ASYMPTOTIC PROPERTIES OF POWER VARIATIONS OF L\'{E}VY PROCESSES Jean Jacod (IMJ) We determine the asymptotic behavior of the realized power variations, or more generally of sums of a given test function evaluated at the successive increments of a L\'{e}vy process. One can completely elucidate the first order behavior (convergence in probability, possibly after normalization). As for the associated CLT, one can show some versions of it, but only in a limited number of cases. In some other cases, a CLT just does not exist. http://front.math.ucdavis.edu/math.PR/0511052 --------------------------------------------------------------- 3855. THE FAIR AND RANDOM MAXIMAL DIVISION OF "PIZZA" Floyd E. Brown and Anant P. Godbole Consider n straight line cuts of a circular pizza made so as to maximize the number of pieces. We investigate how fair such a maximal division may be and how many slices are obtained if the cuts are successfully made with a certain probability. http://front.math.ucdavis.edu/math.PR/0512177 --------------------------------------------------------------- 3856. MULTI-SCALING OF THE $N$-POINT DENSITY FUNCTION FOR COALESCING BROWNIAN MOTIONS R. Munasinghe and R. Rajesh and R. Tribe and O. Zaboronski This paper gives a derivation for the large time asymptotics of the $n $-point density function of a system of coalescing Brownian motions on $\bf{R}$. http://front.math.ucdavis.edu/math.PR/0512179 --------------------------------------------------------------- 3857. STATISTICAL MECHANICAL SYSTEMS ON COMPLETE GRAPHS, INFINITE EXCHANGEABILITY, FINITE EXTENSIONS AND A DISCRETE FINITE MOMENT PROBLEM Thomas Liggett and Jeffrey Steif and Balint Toth We show that a large collection of statistical mechanical systems with quadratically represented Hamiltonians on the complete graph can be extended to infinite exchangeable processes. This includes all ferromagnetic Ising, Potts and Heisenberg models. By de Finetti's theorem, this is equivalent to showing that these probability measures can be expressed as averages of product measures. We provide examples showing that ``ferromagnetism'' is not however in itself sufficient and also study in some detail the Ising model with an additional 3-body interaction. Finally, we study the question of how much the antiferromagnetic Ising model can be extended. In this direction, we obtain sharp asymptotic results via a solution to a new moment problem. We also obtain a ``formula'' for the extension which is valid in many cases. http://front.math.ucdavis.edu/math.PR/0512191 --------------------------------------------------------------- 3858. FELLER PROPERTY AND INFINITESIMAL GENERATOR OF THE EXPLORATION PROCESS Romain Abraham (MAPMO) and Jean-Francois Delmas (CERMICS) We consider the exploration process associated to the continuous random tree (CRT) built using a Levy process with no negative jumps. This process has been studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is a useful tool to study CRT as well as super-Brownian motion with general branching mechanism. In this paper we prove this process is Feller, and we compute its infinitesimal generator on exponential functionals and give the corresponding martingale. http://front.math.ucdavis.edu/math.PR/0512195 --------------------------------------------------------------- 3859. STRICTLY STABLE DISTRIBUTIONS ON CONVEX CONES Youri Davydov and Ilya Molchanov and Sergei Zuyev Using the LePage representation, a strictly stable random element in a Banach space with $\alpha\in(0,2)$ can be represented as a sum of points of a Poisson process. This point process is union-stable, i.e. the union of its two independent copies coincides in distribution with the rescaled original point process. These concepts makes sense in any convex cone, i.e. in a commutative semigroup equipped with multiplication by numbers, and lead to a construction of stable laws in general cones by means of the LePage series. The corresponding limit theorem shows that random samples (or binomial point processes) converge in distribution to the union-stable Poisson point process, and so yields a limit theorem for normalised sums of random elements with $\alpha$-stable limit for $\alpha\in(0,1)$. By using the technique of harmonic analysis on semigroups we characterise distributions of $\alpha$-stable random elements and show how possible values of $\alpha$ relate to the properties of the semigroup and the corresponding scaling operation, in particular, their distributivity properties. The approach developed in the paper not only makes it possible to handle stable distributions in rather general cones (like spaces of sets or measures), but also provides an alternative way to prove classical limit theorems and deduce the LePage representation for strictly stable random vectors in Banach spaces. http://front.math.ucdavis.edu/math.PR/0512196 --------------------------------------------------------------- 3860. ON TIME INHOMOGENEOUS CONTROLLED DIFFUSION PROCESSES IN DOMAINS Hongjie Dong and N.V. Krylov Time inhomogeneous controlled diffusion processes in both cylindrical and non-cylindrical domains are considered. Bellman's principle and its applications to proving the continuity of value functions are investigated. http://front.math.ucdavis.edu/math.PR/0512200 --------------------------------------------------------------- 3861. THE CRITICAL RANDOM GRAPH, WITH MARTINGALES Asaf Nachmias and Yuval Peres We give a short proof that the largest component of the random graph $G(n, 1/n)$ is of size approximately $n^{2/3}$. The proof gives explicit bounds for the probability that the ratio is very large or very small. http://front.math.ucdavis.edu/math.PR/0512201 --------------------------------------------------------------- 3862. BALLS-IN-BINS WITH FEEDBACK AND BROWNIAN MOTION Roberto Oliveira In a balls-in-bins process with feedback, balls are sequentially thrown into bins so that the probability that a bin with n balls obtains the next ball is proportional to f(n) for some function f. A commonly studied case where there are two bins and f(n) = n^p for p > 0, and our goal is to study the fine behavior of this process with two bins and a large initial number t of balls. Perhaps surprisingly, Brownian Motions are an essential part of both our proofs. For p>1/2, it was known that with probability 1 one of the bins will lead the process at all large enough times. We show that if the first bin starts with t+\lambda\sqrt{t} balls (for constant \lambda\in \R), the probability that it always or eventually leads has a non-trivial limit depending on \lambda. For p\leq 1/2, it was known that with probability 1 the bins will alternate in leadership. We show, however, that if the initial fraction of balls in one of the bins is >1/2, the time until it is overtaken by the remaining bin scales like \Theta({t^{1+1/(1-2p)}}) for p<1/2 and \exp(\Theta{t}) for p=1/2. In fact, the overtaking time has a non-trivial distribution around the scaling factors, which we determine explicitly. Our proofs use a continuous-time embedding of the balls-in-bins process (due to Rubin) and a non-standard approximation of the process by Brownian Motion. The techniques presented also extend to more general functions f. http://front.math.ucdavis.edu/math.PR/0510648 --------------------------------------------------------------- 3863. ALMOST SURE ASYMPTOTICS FOR A DIFFUSION PROCESS IN A DRIFTED BROWNIAN POTENTIAL Alexis Devulder (PMA) We study a one-dimensional diffusion process in a drifted Brownian potential. We characterize the upper functions of its hitting times in the sense of Paul L\'evy, and determine the lower limits in terms of an iterated logarithm law. http://front.math.ucdavis.edu/math.PR/0511053 --------------------------------------------------------------- 3864. LARGE DEVIATION PRINCIPLE FOR ENHANCED GAUSSIAN PROCESSES Peter Friz and Nicolas Victoir We study large deviation principles for Gaussian processes lifted to the free nilpotent group of step N. We apply this to a large class of Gaussian processes lifted to geometric rough paths. A large deviation principle for enhanced (fractional) Brownian motion, in Hoelder- or modulus topology, appears as special case. http://front.math.ucdavis.edu/math.PR/0512213 --------------------------------------------------------------- 3865. FELLER PROCESSES ON NON-LOCALLY COMPACT SPACES Tomasz Szarek We introduce the ergodic condition which assures the existence of an invariant measure for Feller processes defined on an arbitrary complete and separable metric space. http://front.math.ucdavis.edu/math.PR/0512221 --------------------------------------------------------------- 3866. TAIL BEHAVIOUR OF MULTIPLE RANDOM INTEGRALS AND U-STATISTICS Peter Major This paper contains sharp estimates about the distribution of multiple random integrals of functions of several variables with respect to a normalized empirical measure, about the distribution of U-statistics and multiple Wiener-Ito integrals with respect to a white noise. It also contains good estimates about the supremum of appropriate classes of such integrals or U-statistics. The proof of most results is omitted, I have concentrated on the explanation of their content and the picture behind them. I also tried to explain the reason for the investigation of such questions. My goal was to yield such a presentation of the results which a non-expert also can understand, and not only on a formal level. http://front.math.ucdavis.edu/math.PR/0512238 --------------------------------------------------------------- 3867. CRITICAL SCALING FOR THE SIMPLE SIS STOCHASTIC EPIDEMIC R. G. Dolgoarshinnykh Steven P. Lalley We exhibit a scaling law for the critical SIS stochastic epidemic: If at time 0 the population consists of square root N infected and N - square root N susceptible individuals, then when time and number currently infected are both scaled by square root N, the resulting process converges, for large N, to a diffusion process related to the Feller diffusion by a change of drift. As a consequence, the rescaled size of the epidemic has a limit law that coincides with that of a first-passage time for the standard Ornstein- Uhlenbeck process. These results are the analogues for the SIS epidemic of results of Martin-Lof for the simple SIR epidemic. http://front.math.ucdavis.edu/math.PR/0512252 --------------------------------------------------------------- 3868. STRONG SOLUTIONS OF STOCHASTIC GENERALIZED POROUS MEDIA EQUATIONS: EXISTENCE, UNIQUENESS AND ERGODICITY Giuseppe Da Prato and Boris L. Rozovskii and Michael R\"ockner and Feng-Yu Wang Explicit conditions are presented for the existence, uniqueness and ergodicity of the strong solution to a class of generalized stochastic porous media equations. Our estimate of the convergence rate is sharp according to the known optimal decay for the solution of the classical (deterministic) porous medium equation. http://front.math.ucdavis.edu/math.PR/0512259 --------------------------------------------------------------- 3869. HARMONIC CONTINUOUS TIME BRANCHING MOMENTS Didier Piau We show that the mean inverse populations of nondecreasing, square integrable, continuous time branching processes decrease to zero like the inverse of their mean population if and only if the initial population k is greater than a threshold m, which is at least one. If furthermore k is greater than a second threshold m', which is at least m, the normalized mean inverse population is at most 1/(k-m'). We express m and m' as explicit functionals of the reproducing distribution, we discuss some analogues for discrete time branching processes, and we link these results to the behavior of random products involving i.i.d. nonnegative sums. http://front.math.ucdavis.edu/math.PR/0511058 --------------------------------------------------------------- 3870. GLOBAL REGULARITY AND BOUNDS FOR SOLUTIONS OF PARABOLIC EQUATIONS FOR PROBABILITY MEASURES Vladimir I. Bogachev and Michael R\"ockner and Stanislav V. Shaposhnikov Given a second order parabolic operator $$ Lu(t,x) :=\frac{\partial u(t,x)}{\partial t} + a^{ij}(t,x)\partial_{x_i}\partial_{x_j}u(t,x) + b^i(t,x)\partial_{x_i}u(t,x), $$ we consider the weak parabolic equation $L^{*}\mu=0$ for Borel probability measures on $(0,1)\times\mathbb{R}^d$. The equation is understood as the equality $$ \int_{(0,1)\times\mathbb{R}^d} Lu d\mu =0 $$ for all smooth functions $u$ with compact support in~$(0,1)\times\mathbb{R}^d$. This equation is satisfied for the transition probabilities of the diffusion process associated with~$L$. We show that under broad assumptions $\mu$ has the form $\mu= \varrho(t,x) dt dx$, where the function $x\mapsto \varrho(t,x)$ is Sobolev, $|\nabla_x \varrho(x,t)|^2/\varrho(t,x)$ is Lebesgue integrable over $[0,\tau]\times\mathbb{R}^d$, and $\varrho\in L^p([0,\tau]\times \mathbb{R}^d)$ for all $p\in [1,+\infty)$ and $\tau<1$. Moreover, a sufficient condition for the uniform boundedness of $\varrho$ on $[0,\tau]\times\mathbb{R}^d$ is given. http://front.math.ucdavis.edu/math.PR/0512264 --------------------------------------------------------------- 3871. CHAOTIC STATES AND STOCHASTIC INTEGRATION IN QUANTUM SYSTEMS V. P. Belavkin Quantum chaotic states over a noncommutative monoid, a unitalization of a noncommutative Ito algebra parametrizing a quantum stochastic Levy process, are described in terms of their infinitely divisible generating functionals over the monoid-valued processes on an atomless `space-time' set. A canonical decomposition of the logarithmic conditionally posive-definite generating functional is constructed in a pseudo-Euclidean space, given by a quadruple defining the monoid triangular operator representation and a cyclic zero pseudo-norm state in this space. It is shown that the exponential representation in the corresponding pseudo-Fock space yields the infinitely-divisible generating functional with respect to the exponential state vector, and its compression to the Fock space defines the cyclic infinitly-divisible representation associated with the Fock vacuum state. The structure of states on an arbitrary Ito algebra is studied with two canonical examples of quantum Wiener and Poisson states. A generalized quantum stochastic nonadapted multiple integral is explicitly defined in Fock scale, its continuity and quantum stochastic differentiability is proved. A unified non-adapted and functional quantum Ito formula is discovered and established both in weak and strong sense, and the multiplication formula on the exponential Ito algebra is found for the relatively bounded kernel- operators in Fock scale. The unitarity and projectivity properties of nonadapted quantum stochastic linear differential equations are studied, and their solution is constructed for the locally bounded nonadapted generators in terms of the chronological products in the underlying kernel algebra canonically represented by triangular operators in the pseudo-Fock space. http://front.math.ucdavis.edu/math.PR/0512265 --------------------------------------------------------------- 3872. WEAK SOLUTIONS TO THE STOCHASTIC POROUS MEDIA EQUATION VIA KOLMOGOROV Viorel Barbu and Vladimir I. Bogachev and Giuseppe Da Prato and Michael R\"ockner A stochastic version of the porous medium equation with coloured noise is studied. The corresponding Kolmogorov equation is solved in the space $L^2(H,\nu)$ where $\nu$ is an infinitesimally excessive measure. Then a weak solution is constructed. http://front.math.ucdavis.edu/math.PR/0512266 --------------------------------------------------------------- 3873. EXPLICIT FORMULAS FOR THE MOMENTS OF THE SOJOURN TIME IN THE M/ G/1 PROCESSOR SHARING QUEUE WITH PERMANENT JOBS S.F.Yashkov We give some representation about recent achievements in analysis of the M/G/1 queue with egalitarian processor sharing discipline (EPS). The new formmulas are derived for the j-th moments (j=1,2,...) of the (conditional) stationary sojourn time in the M/G/1--EPS queue with K (K=0,1,2,...) permanent jobs of infinite size. We discuss also how to simplify the computations of the moments. http://front.math.ucdavis.edu/math.PR/0512281 --------------------------------------------------------------- 3874. A PREDICTIVE THEORY OF GAMES David H. Wolpert Conventional noncooperative game theory hypothesizes that the joint strategy of a set of players in a game must satisfy an "equilibrium concept". All other joint strategies are considered impossible; the only issue is what equilibrium concept is "correct". This hypothesis violates the desiderata underlying probability theory. Indeed, probability theory renders moot the problem of what equilibrium concept is correct - every joint strategy can arise with non-zero probability. Rather than a first-principles derivation of an equilibrium concept, game theory requires a first-principles derivation of a distribution over joint (mixed) strategies. This paper shows how information theory can provide such a distribution over joint strategies. If a scientist external to the game wants to distill such a distribution to a point prediction, that prediction should be set by decision theory, using their (!) loss function. So the predicted joint strategy - the "equilibrium concept" - varies with the external scientist's loss function. It is shown here that in many games, having a probability distribution with support restricted to Nash equilibria - as stipulated by conventional game theory - is impossible. It is also show how to: i) Derive an information-theoretic quantification of a player's degree of rationality; ii) Derive bounded rationality as a cost of computation; iii) Elaborate the close formal relationship between game theory and statistical physics; iv) Use this relationship to extend game theory to allow stochastically varying numbers of players. http://front.math.ucdavis.edu/nlin.AO/0512015 --------------------------------------------------------------- 3875. INFINITE DIMENSIONAL ITO ALGEBRAS OF QUANTUM WHITE NOISE V. P. Belavkin A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Ito B*- algebra, generalizing the C*-algebra is defined to include the Banach infinite dimensional Ito algebras of quantum Brownian and quantum Levy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. It is proved that every Ito algebra is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy- Khinchin theorem in the classical case. http://front.math.ucdavis.edu/math.PR/0512288 --------------------------------------------------------------- 3876. POSITIVE DEFINITE GERMS OF QUANTUM STOCHASTIC PROCESSES V. P. Belavkin A new notion of stochastic germs for quantum processes is introduced and a characterisation of the stochastic differentials for positive definite (PD) processes is found in terms of their germs for arbitrary Ito algebra. A representation theorem, giving the pseudo-Hilbert dilation for the germ-matrix of the differential, is proved. This suggests the general form of quantum stochastic evolution equations with respect to the Poisson (jumps), Wiener (diffusion) or general quantum noise. http://front.math.ucdavis.edu/math.PR/0512289 --------------------------------------------------------------- 3877. ON STOCHASTIC GENERATORS OF POSITIVE DEFINITE EXPONENTS V. P. Belavkin A characterisation of quantum stochastic positive definite (PD) exponent is given in terms of the conditional positive definiteness (CPD) of their form-generator. The pseudo-Hilbert dilation of the stochastic form- generator and the pre-Hilbert dilation of the corresponding dissipator is found. The structure of quasi-Poisson stochastic generators giving rise to a quantum stochastic birth processes is studied. http://front.math.ucdavis.edu/math.PR/0512290 --------------------------------------------------------------- 3878. POISSON KERNEL AND GREEN FUNCTION OF THE BALL IN REAL HYPERBOLIC SPACES T. Byczkowski and J. Malecki Let $(X_t)_{t\geq0}$ be the $n$-dimensional hyperbolic Brownian motion, that is the diffusion on the real hyperbolic space $\D^n$ having the Laplace-Beltrami operator as its generator. The aim of the paper is to derive the formulas for the Gegenbauer transform of the Poisson kernel and the Green function of the ball for the process $(X_t)_{t\geq0}$. Under some additional hypotheses we give the formulas for the Poisson kernel itself. In particular, we provide formulas in $\D^4$ and $\D^6$ spaces for the Poisson kernel and the Green function as well. http://front.math.ucdavis.edu/math.PR/0512294 --------------------------------------------------------------- 3879. RANDOM HOMEOMORPHISMS AND FOURIER EXPANSIONS - THE POINTWISE BEHAVIOR Gady Kozma Let phi be a Dubins-Freedman random homeomorphism on [0,1] derived from the base measure uniform on the vertical line x=1/2, and let f be a periodic function satisfying that |f(x)-f(0)| = o(1/log log log 1/x). Then the Fourier expansion of f composed with phi converges at 0 with probability 1. In the condition on f, o cannot be replaced by O. Also we deduce some 0-1 laws for this kind of problems. http://front.math.ucdavis.edu/math.CA/0511036 --------------------------------------------------------------- 3880. BINOMIAL UPPER BOUNDS ON GENERALIZED MOMENTS AND TAIL PROBABILITIES OF (SUPER)MARTINGALES WITH DIFFERENCES BOUNDED FROM ABOVE Iosif Pinelis Let (S_0,S_1,...) be a supermartingale relative to a nondecreasing sequence of sigma-algebras H_0,H_1,..., with S_0\le0 almost surely (a.s.) and differences X_i:=S_i-S_{i-1}. Suppose that X_i\le d and Var(X_i|H_ {i-1})\le \si_i^2 a.s. for every i=1,2,..., where d>0 and \si_i>0 are non-random constants. Let T_n:=Z_1+...+Z_n, where Z_1,...,Z_n are i.i.d. r.v.'s each taking on only two values, one of which is d, and satisfying the conditions E(Z_i)=0 and Var(Z_i)=\si^2:=(\si_1^2+...+\si_n^2)/n. Then, based on a comparison inequality between generalized moments of S_n and T_n for a rich class of generalized moment functions, the tail comparison inequality P(S_n \ge y) \le c P^{\lin,\lc}(T_n \ge y+h/2)\quad\forall y\in\R is obtained, where c:=e^2/2=3.694..., h:=d+\si^2/d, and the function y\mapsto P^{\lin, \lc}(T_n > y) is the least log-concave majorant of the linear interpolation of the tail function y\mapsto P(T_n \ge y) over the lattice of all points of the form nd+kh (k\in\Z). An explicit formula for P^{\lin,\lc}(T_n\ge y+h/2) is given. Another, similar bound is given under somewhat different conditions. It is shown that these bounds improve significantly upon known bounds. http://front.math.ucdavis.edu/math.PR/0512301 --------------------------------------------------------------- 3881. LOCAL STRUCTURE OF RANDOM QUADRANGULATIONS Maxim Krikun (IEC) This paper is an adaptation of a method used in math.PR/0311127 to the model of random quadrangulations. We prove local weak convergence of uniform measures on quadrangulations and show that local growth of quadrangulation is governed by certain critical time-reversed branching process. As an intermediate result we calculate a biparametric generating function for certain class of quadrangulations with boundary. http://front.math.ucdavis.edu/math.PR/0512304 --------------------------------------------------------------- 3882. LARGE SYSTEMS OF PATH-REPELLENT BROWNIAN MOTIONS IN A TRAP AT POSITIVE TEMPERATURE Stefan Adams and Jean-Bernard Bru and Wolfgang Koenig We study a model of $ N $ mutually repellent Brownian motions under confinement to stay in some bounded region of space. Our model is defined in terms of a transformed path measure under a trap Hamiltonian, which prevents the motions from escaping to infinity, and a pair-interaction Hamiltonian, which imposes a repellency of the $N$ paths. In fact, this interaction is an $N$-dependent regularisation of the Brownian intersection local times, an object which is of independent interest in the theory of stochastic processes. The time horizon (interpreted as the inverse temperature) is kept fixed. We analyse the model for diverging number of Brownian motions in terms of a large deviation principle. The resulting variational formula is the positive-temperature analogue of the well-known Gross-Pitaevskii formula, which approximates the ground state of a certain dilute large quantum system; the kinetic energy term of that formula is replaced by a probabilistic energy functional. This study is a continuation of the analysis in \cite{ABK04} where we considered the limit of diverging time (i.e., the zero-temperature limit) with fixed number of Brownian motions, followed by the limit for diverging number of motions. \bibitem[ABK04]{ABK04} {\sc S.~Adams, J.-B.~Bru} and {\sc W.~K \"onig}, \newblock Large deviations for trapped interacting Brownian particles and paths, \newblock {\it Ann. Probab.}, to appear (2004). http://front.math.ucdavis.edu/math.PR/0512305 --------------------------------------------------------------- 3883. A FUNCTIONAL CENTRAL LIMIT THEOREM FOR A CLASS OF URN MODELS Gopal K Basak and Amites Dasgupta We construct an independent increments Gaussian process associated to a class of multicolor urn models. The construction uses random variables from the urn model which are different from the random variables for which central limit theorems are available in the two color case. http://front.math.ucdavis.edu/math.PR/0512325 --------------------------------------------------------------- 3884. COUPLING ALL THE LEVY STOCHASTIC AREAS OF MULTIDIMENSIONAL BROWNIAN MOTION Wilfrid Kendall It is shown how to construct a successful co-adapted coupling of two copies of an n-dimensional Brownian motion while simultaneously coupling all corresponding copies of Levy stochastic areas. It is conjectured that successful co-adapted couplings still exist when the Levy stochastic areas are replaced by a finite set of multiply-iterated path-and-time integrals, subject to algebraic compatibility of the initial conditions. http://front.math.ucdavis.edu/math.PR/0512336 --------------------------------------------------------------- 3885. QUANTUM STOCHASTIC SEMIGROUPS AND THEIR GENERATORS V. P. Belavkin A rigged Hilbert space characterisation of the unbounded generators of quantum completely positive (CP) stochastic semigroups is given. The general form and the dilation of the stochastic completely dissipative (CD) equation over the algebra L(H) is described, as well as the unitary quantum stochastic dilation of the subfiltering and contractive flows with unbounded generators is constructed. http://front.math.ucdavis.edu/math.PR/0512360 --------------------------------------------------------------- 3886. QUANTUM STOCHASTIC CALCULUS AND QUANTUM NONLINEAR FILTERING V. P. Belavkin A *-algebraic indefinite structure of quantum stochastic (QS) calculus is introduced and a continuity property of generalized nonadapted QS integrals is proved under the natural integrability conditions in an infinitely dimensional nuclear space. The class of nondemolition output QS processes in quantum open systems is characterized in terms of the QS calculus, and the problem of QS nonlinear filtering with respect to nondemolition continuous measurments is investigated. The stochastic calculus of a posteriori conditional expectations in quantum observed systems is developed and a general quantum filtering stochastic equation for a QS process is derived. An application to the description of the spontaneous collapse of the quantum spin under continuous observation is given. http://front.math.ucdavis.edu/math.PR/0512362 --------------------------------------------------------------- 3887. LOGARITHMIC ASYMPTOTICS FOR THE NUMBER OF PERIODIC ORBITS OF THE TEICHMUELLER FLOW ON VEECH'S SPACE OF ZIPPERED RECTANGLES Alexander I. Bufetov The logarithmic asymptotics is computed for the growth of the number of periodic orbits for the Teichmueller flow on Veech's moduli space of zippered rectangles. The rate is equal to the entropy of the flow with respect to the absolutely continuous invariant measure. http://front.math.ucdavis.edu/math.DS/0511035 --------------------------------------------------------------- 3888. LOCALIZATION TRANSITION FOR A COPOLYMER IN AN EMULSION F den Hollander and S G Whittington In this paper we study a two-dimensional directed self-avoiding walk model of a random copolymer in a random emulsion. http://front.math.ucdavis.edu/math.PR/0512374 --------------------------------------------------------------- 3889. GIBBS DISTRIBUTIONS FOR RANDOM PARTITIONS GENERATED BY A FRAGMENTATION PROCESS Nathanael Berestycki (U.B.C.) and Jim Pitman (U.C. BERKELEY) In this paper we study random partitions of {1,...,n} where every cluster of size j can be in any of w(j) possible internal states. The Gibbs (n,k,w) distribution is obtained by sampling uniformly among such partitions with k clusters. Gibbs distributions arise naturally as equilibrium distributions of reversible coagulation - fragmentation processes. The goal of this work is to study random processes where at step k the process has the Gibbs (n,k,w) distribution, so that this microscopical equilibrium is subject to irreversible fragmentation as time evolves. It is not always possible to combine those two features, and in our main result we identify those weight sequences w (j) for which such a process exists subject to some simplifying assumptions. In this case the time-reversed process turns out to be the discrete Marcus- Lushnikov coalescent process with affine collision rate K(x,y)=a+b(x+y) for some real numbers a and b. http://front.math.ucdavis.edu/math.PR/0512378 --------------------------------------------------------------- 3890. A QUANTITATIVE INVESTIGATION INTO THE ACCUMULATION OF ROUNDING ERRORS IN THE NUMERICAL SOLUTION OF ODES Sebastian Mosbach and Amanda G. Turner We examine numerical rounding errors of some deterministic solvers for systems of ordinary differential equations (ODEs). We show that the accumulation of rounding errors results in a solution that is inherently random and we obtain the theoretical distribution of the trajectory as a function of time, the step size and the numerical precision of the computer. We consider, in particular, systems which amplify the effect of the rounding errors so that over long time periods the solutions exhibit divergent behaviour. By performing multiple repetitions with different values of the time step size, we observe numerically the random distributions predicted theoretically. We mainly focus on the explicit Euler and RK4 methods but also briefly consider more complex algorithms such as the implicit solvers VODE and RADAU5. http://front.math.ucdavis.edu/math.NA/0512364 --------------------------------------------------------------- 3891. NORMAL DOMINATION OF (SUPER)MARTINGALES Iosif Pinelis Let (S_0,S_1,...) be a supermartingale relative to a nondecreasing sequence of \sigma-algebras (H_{\le0},H_{\le1},...), with S_0\le0 almost surely (a.s.) and differences X_i:=S_i-S_{i-1}. Suppose that for every i=1,2,... there exist H_{\le(i-1)}-measurable r.v.'s C_{i-1} and D_{i-1} and a positive real number s_i such that C_{i-1}\le X_i\le D_{i-1} and D_{i-1}-C_{i-1}\le 2 s_i a.s. Then for all real t and natural n one has \E f_t(S_n)\le\E f_t(sZ), where f_t(x):=\max(0,x-t)^5, s:=\sqrt{s_1^2+...+s_n^2}, and Z is N(0,1). In particular, this implies P(S_n\ge x)\le c_{5,0}P(Z\ge x/s) for all x in \R, where c_{5,0}=5!(e/5)^5=5.699.... Results for \max_{0\le k\le n}S_k in place of S_n and for concentration of measure also follow. http://front.math.ucdavis.edu/math.PR/0512382 --------------------------------------------------------------- 3892. RELATIVE ENTROPY AND WAITING TIMES FOR CONTINUOUS-TIME MARKOV PROCESSES Jean-Rene Chazottes and Cristian Giardina and Frank Redig For discrete-time stochastic processes, there is a close connection between return/waiting times and entropy. Such a connection cannot be straightforwardly extended to the continuous-time setting. Contrarily to the discrete- time case one does need a reference measure and so the natural object is relative entropy rather than entropy. In this paper we elaborate on this in the case of continuous-time Markov processes with finite state space. A reference measure of special interest is the one associated to the time-reversed process. In that case relative entropy is interpreted as the entropy production rate. The main results of this paper are: almost-sure convergence to relative entropy of suitable waiting-times and their fluctuation properties (central limit theorem and large deviation principle). http://front.math.ucdavis.edu/math.PR/0512386 --------------------------------------------------------------- 3893. ASYMPTOTIC DIRECTION FOR RANDOM WALKS IN RANDOM ENVIRONMENTS Fran\c{c}ois Simenhaus (PMA) In this paper we study the property of asymptotic direction for random walks in random i.i.d. environments (RWRE). We prove that if the set of directions where the walk is transient is non empty and open, the walk admits an asymptotic direction. The main tool to obtain this result is the construction of a renewal structure with cones. We also prove that RWRE admits at most two opposite asymptotic directions. http://front.math.ucdavis.edu/math.PR/0512388 --------------------------------------------------------------- 3894. RANDOMLY GROWING BRAID ON THREE STRANDS AND THE MANTA RAY, WITH APPENDIX Jean Mairesse and Fr\'ed\'eric Math\'eus Consider the braid group B3 = < a,b | aba = bab > and the nearest neighbor random walk defined by a probability \nu with support {a,b,a^-1,b^-1}. The rate of escape of the walk is explicitely expressed in function of the unique solution of a set of eight polynomial equations of degree three over eight indeterminates. We also explicitely describe the harmonic measure of the induced random walk on B3 quotiented by its center. The method and results apply, mutatis mutandis, to nearest neighbor random walks on dihedral Artin groups. http://front.math.ucdavis.edu/math.PR/0512391 --------------------------------------------------------------- 3895. A UNIVERSAL DILATION OF DISCRETE MARKOV EVOLUTIONS M. Gregoratti Given a finite state space E, we build a universal dilation for all possible discrete time Markov chains on E, homogeneous or not: we introduce a second system (an ``environment'') and a deterministic invertible time- homogeneous global evolution of the system E with this environment such that any Markov evolution of E can be realized by a proper choice of the initial (random) state of the environment, which therefore determines the transition probabilities of the system. We also compare this dilation with the quantum dilations of a Quantum Dynamical Semigroup: given a Classical Markov Semigroup, we show that it can be extended to a Quantum Dynamical Semigroup for which we can find a quantum dilation to a group of *-automorphisms admitting an invariant abelian subalgebra where this quantum dilation gives just our classical dilation. http://front.math.ucdavis.edu/math.PR/0512393 --------------------------------------------------------------- 3896. LARGE DEVIATIONS OF THE EMPIRICAL CURRENT IN INTERACTING PARTICLE SYSTEMS L. Bertini and A. De Sole and D. Gabrielli and G. Jona-Lasinio and C. Landim We study current fluctuations in lattice gases in the hydrodynamic scaling limit. More precisely, we prove a large deviation principle for the empirical current in the symmetric simple exclusion process with rate functional I. We then estimate the asymptotic probability of a fluctuation of the average current over a large time interval and show that the corresponding rate function can be obtained by solving a variational problem for the functional I. For the symmetric simple exclusion process the minimizer is time independent so that this variational problem can be reduced to a time independent one. On the other hand, for other models the minimizer is time dependent. This phenomenon is naturally interpreted as a dynamical phase transition. http://front.math.ucdavis.edu/math.PR/0512394 --------------------------------------------------------------- 3897. CONFORMAL INVARIANCE OF ISORADIAL DIMER MODELS & THE CASE OF TRIANGULAR QUADRI-TILINGS B. de Tili\`ere We consider dimer models on graphs which are bipartite, periodic and satisfy a geometric condition called {\em isoradiality}, defined in \cite {Kenyon3}. We show that the scaling limit of the height function of any such dimer model is $1/\sqrt{\pi}$ times a Gaussian free field. Triangular quadri-tilings were introduced in \cite{Bea}; they are dimer models on a family of isoradial graphs arising form rhombus tilings. By means of two height functions, they can be interpreted as random interfaces in dimension 2+2. We show that the scaling limit of each of the two height functions is $1/\sqrt{\pi}$ times a Gaussian free field, and that the two Gaussian free fields are independent. http://front.math.ucdavis.edu/math.PR/0512395 --------------------------------------------------------------- 3898. THE MONOTONICITY CONDITION FOR BSDE ON MANIFOLDS Fabrice Blache (IAM) In two preceding articles, we studied the problem of the existence and uniqueness of a solution to some general BSDE on manifolds. In these two articles, we assumed some Lipschitz conditions on the drift $f(b,x,z) $. The purpose of this article is to extend the existence and uniqueness results under weaker assumptions, in particular a monotonicity condition in the variable $x$. This extends well-known results for Euclidean BSDE. http://front.math.ucdavis.edu/math.PR/0512403 --------------------------------------------------------------- 3899. OPERATOR MARKOVIAN COCYCLES VIA ASSOCIATED SEMIGROUPS J. Martin Lindsay and Stephen J. Wills A recent characterisation of Fock-adapted contraction operator stochastic cocycles on a Hilbert space, in terms of their associated semigroups, yields a general principle for the construction of such cocycles by approximation of their stochastic generators. This leads to new existence results for quantum stochastic differential equations. We also give necessary and sufficient conditions for a cocycle to satisfy such an equation. http://front.math.ucdavis.edu/math.FA/0512398 --------------------------------------------------------------- 3900. MARKOV MEASURES ON YOUNG TABLEAUX AND INDUCED REPRESENTATIONS ON THE INFINITE SYMMETRIC GROUP A.M.Vershik and N.V.Tsilevich We show that the class of inductive limits of the representations of finite symmetric groups with simple spectrum coinsides with the class of Markov representations of the infinite symmetric group associated with Markov measures on the space of infinite Young tableaux. We also show that the representations of infinite symmetric group induced from identity representation of two-block Young subgroup are Markov representations and find explicit formulas for transition probabilities of corresponding Markov measure on the Young diagrmas. Induced two-row representations of finite symmetric group are studied using tensor model of those representations which alows easily to obtain the formulas for Gel'fand-Zetlin basis. http://front.math.ucdavis.edu/math.RT/0512389 --------------------------------------------------------------- 3901. RECONSTRUCTION THEOREM FOR QUANTUM STOCHASTIC PROCESSES V. P. Belavkin Statistically interpretable axioms are formulated that define a quantum stochastic process (QSP) as a causally ordered operator field in an arbitrary space-time region T of an open quantum system under a sequential observation at a discrete space-time localization. It is shown that to every QSP described in the weak sense by a self-consistent system of causally ordered correlation kernels there corresponds a unique, up to unitary equivalence, minimal QSP in the strong sense. It is shown that the proposed QSP construction, which reduces in the case of the linearly ordered discrete T=Z to the construction of the inductive limit of Lindblad's canonical representations, corresponds to Kolmogorov's classical reconstruction if the order on T is ignored and leads to Lewis construction if one uses the system of all (not only causal) correlation kernels, regarding this system as lexicographically preordered on T. The approach presented encompasses both nonrelativistic and relativistic irreversible dynamics of open quantum systems and fields satisfying the conditions of local commutativity and semigroup covariance. Also given are necessary and sufficient conditions of dynamicity (or conditional Markovianity) and regularity, these leading to the properties of complete mixing (relaxation) and ergodicity of the QSP. http://front.math.ucdavis.edu/math.PR/0512410 --------------------------------------------------------------- 3902. SEMILOGICS, QUASILOGICS AND OTHER QUANTUM STRUCTURES V. P. Belavkin We give an axiomatic formulation of quantum structures like semilogics and quasilogics which generalize the boolean semirings of events and fuzzy logics. The notions of distributions, states, representations observables and semiobservables are introduced and their Hilbert space realizations are found. The closed and open structures in semilogics are introduced and the regular distributions on the semilogics are studied. http://front.math.ucdavis.edu/math.PR/0512413 --------------------------------------------------------------- 3903. OCCUPATION TIME FLUCTUATIONS OF POISSON AND EQUILIBRIUM FINITE VARIANCE BRANCHING SYSTEMS Piotr Milos Functional limit theorems are presented for the rescaled occupation time fluctuations process of a critical finite variance branching particle system in $R^d$ with symmetric a-stable motion starting off from either a standard Poisson random field or from the equilibrium distribution for intermediate dimensions a0 and let \{N(t); t\geq 0\} be a counting process independent of the X_i's. For any fixed t\geq 0, define T_{N(t)}:=\frac{X_1^2 + X_2^2 + ... + X_{N(t)} ^2} {(X_1 + X_2 + ... + X_{N(t)})^2} if N(t)\geq 1 and T_{N(t)}:=0 otherwise. We derive limiting distributions for T_{N(t)} by assuming some convergence properties for the counting process. This is even achieved when both the numerator and the denominator defining T_{N(t)} exhibit an erratic behavior (\mathbb{E}X_1=\infty) or when only the numerator has an erratic behavior (\mathbb{E}X_1<\infty and \mathbb{E}X_1^2=\infty). Thanks to these results, we obtain asymptotic properties pertaining to both the sample coefficient of variation and the sample dispersion. http://front.math.ucdavis.edu/math.PR/0511082 --------------------------------------------------------------- 3911. PLONGEMENT STOCHASTIQUE DES SYST\`{E}MES LAGRANGIENS Jacky Cresson (LM-Besan\c{c}on) and S\'{e}bastien Darses (LM-Besan\c {c}on) We define an operator which extends classical differentiation from smooth deterministic functions to certain stochastic processes. Based on this operator, we define a procedure which associates a stochastic analog to standard differential operators and ordinary differential equations. We call this procedure stochastic embedding. By embedding lagrangian systems, we obtain a stochastic Euler-Lagrange equation which, in the case of natural lagrangian systems, is called the embedded Newton equation. This equation contains the stochastic Newton equation introduced by Nelson in his dynamical theory of brownian diffusions. Finally, we consider a diffusion with a gradient drift, a constant diffusion coefficient and having a probability density function. We prove that a necessary condition for this diffusion to solve the embedded Newton equation is that its density be the square of the modulus of a wave function solution of a linear Schr\"{o}dinger equation. http://front.math.ucdavis.edu/math.PR/0510655 --------------------------------------------------------------- 3912. A MOMENTUM CONSERVING MODEL WITH ANOMALOUS THERMAL CONDUCTIVITY IN LOW DIMENSION Giada Basile (CEREMADE) and Cedric Bernardin (UMPA-ENSL) and Stefano Olla (CEREMADE) Anomalous large thermal conductivity has been observed numerically and experimentally in one and two dimensional systems. All explicitly solvable microscopic models proposed until now did not explain this phenomenon and there is an open debate about the role of conservation of momentum. We introduce a model whose thermal conductivity diverges in dimension 1 and 2, while it remains finite in dimension 3. We compute the finite-size thermal conductivity of a system of harmonic oscillators perturbed by a non-linear stochastic dynamics conserving momentum and energy. In the limit as the size N of the system goes to infinity, conductivity diverges like N in dimension 1 and like ln N in dimension 2. Conductivity remains finite if d=3 or if a pinning (on site potential) is present. This result clarify the role of conservation of momentum in the anomalous thermal conductivity. http://front.math.ucdavis.edu/cond-mat/0509688 --------------------------------------------------------------- 3913. MISMATCHED CODEBOOKS AND THE ROLE OF ENTROPY-CODING IN LOSSY DATA COMPRESSION Ioannis Kontoyiannis (Athens U of Econ & Business) and Rami Zamir (Tel-Aviv University) We introduce a universal quantization scheme based on random coding, and we analyze its performance. This scheme consists of a source-independent random codebook (typically_mismatched_ to the source distribution), followed by optimal entropy-coding that is_matched_ to the quantized codeword distribution. A single-letter formula is derived for the rate achieved by this scheme at a given distortion, in the limit of large codebook dimension. The rate reduction due to entropy-coding is quantified, and it is shown that it can be arbitrarily large. In the special case of "almost uniform" codebooks (e.g., an i.i.d. Gaussian codebook with large variance) and difference distortion measures, a novel connection is drawn between the compression achieved by the present scheme and the performance of "universal" entropy-coded dithered lattice quantizers. This connection generalizes the "half-a-bit" bound on the redundancy of dithered lattice quantizers. Moreover, it demonstrates a strong notion of universality where a single "almost uniform" codebook is near-optimal for_any_ source and_any_ difference distortion measure. http://front.math.ucdavis.edu/cs.IT/0511009 --------------------------------------------------------------- 3914. THE K-CORE AND BRANCHING PROCESSES Oliver Riordan The k-core of a graph G is the maximal subgraph of G having minimum degree at least k. In 1996, Pittel, Spencer and Wormald found the threshold $ \lambda_c$ for the emergence of a non-trivial k-core in the random graph $G(n, \lambda/n)$, and the asymptotic size of the k-core above the threshold. We give a new proof of this result using a local coupling of the graph to a suitable branching process. This proof extends to a general model of inhomogeneous random graphs with independence between the edges. As an example, we study the k- core in a certain power-law or `scale-free' graph with a parameter c controlling the overall density of edges. For each k at least 3, we find the threshold value of c at which the k-core emerges, and the fraction of vertices in the k- core when c is \epsilon above the threshold. In contrast to $G(n,\lambda/n)$, this fraction tends to 0 as \epsilon tends to 0. http://front.math.ucdavis.edu/math.CO/0511093 --------------------------------------------------------------- 3915. LIMITING LAWS FOR LONG BROWNIAN BRIDGES PERTURBED BY THEIR ONE- SIDED MAXIMUM, III Bernard Roynette (IEC) and Pierre Vallois (IEC) and Marc Yor (PMA) Results of penalization of a one-dimensional Brownian motion $(X_t) $, by its one-sided maximum $\dis (S_t=\sup_{0 \leq u \leq t}X_u)$, which were recently obtained by the authors are improved with the consideration-in the present paper- of the asymptotic behaviour of the likewise penalized Brownian bridges of length $t$, as $t\to \infty$, or penalizations by functions of $ (S_t,X_t)$, and also the study of the speed of convergence, as $t\to \infty$, of the penalized distributions at time $t$. http://front.math.ucdavis.edu/math.PR/0511102 --------------------------------------------------------------- 3916. EXERCISE REGIONS AND CONTINUITY CORRECTIONS FOR (PERPETUAL) AMERICAN AND BERMUDAN OPTIONS ON MULTIPLE ASSETS Frederik S Herzberg In a general Markovian martingale framework for multi-dimensional options, the existence of optimal exercise regions for multi-dimensional Bermudan options is established. Afterwards one can proceed to prove explicit formulae and asymptotic results on the perpetual American-Bermudan (barrier) put option price difference (``continuity correction'') when the argument of this function -- taken to be the (logarithmic) start price -- approaches the exercise boundary. In particular, results of Feller's shall be generalised to show that an extrapolation from the exact Bermudan prices to the American price cannot be polynomial in the exercise mesh size in the setting of many common market models, and more specific bounds on the natural scaling exponent of the non-polynomial extrapolation for a number of (both one- and multi- dimensional) market models will be deduced. http://front.math.ucdavis.edu/math.PR/0511106 --------------------------------------------------------------- 3917. RUIN ANALYSIS IN CONSTANT ELASTICITY OF VARIANCE MODEL WITH LARGE INITIAL FUNDS F. Klebaner and R. Liptser We consider the value process described by the Constant Elasticity of Variance Model (CEV), given by the stochastic differential equation $$ dX_t=\alpha X_tdt+\sigma X^\gamma_tdB_t, $$ with $X_0=K$, and $1/2\le \gamma<1$. Denote the time of ruin $\tau_K=\inf\{t:X_t=0\}$. We give an asymptotic for the ruin probability by time $T$, $\mathsf{P}(\tau_K \le T)$ \begin{gather*} \lim\limits_{K\to\infty} \frac{1}{K^{2(1-\gamma)}}\log\mathsf{P}(\tau_K\le T) =-\begin{cases} \frac{\alpha}{\sigma^2[1-e^{-2\alpha(1-\gamma)T}]}, & \alpha\ne 0 \ \frac{1}{2\sigma^2(1-\gamma)T}, & \alpha=0 \end{cases}. \end{gather*} The most likely paths to ruin is also found. The results are obtained by solving a control problem arising with help the Large Deviations Principle (LDP). http://front.math.ucdavis.edu/math.PR/0511116 --------------------------------------------------------------- 3918. LEMME DE COHERENCE ET TH\'{E}OR\`{E}ME DE NOETHER STOCHASTIQUE Jacky Cresson (LM-Besan\c{c}on) and S\'{e}bastien Darses (LM-Besan\c {c}on) The stochastic embedding procedure associates a stochastic Euler- Lagrange equation (SEL) to the standard Euler-Lagrange equation (EL). Can we derive (SEL) from a generalized least action principle? To address this question, we develop a stochastic calculus of variation initiated by Yasue. We give a stochastic analog F of the lagrangian action functional. We introduce a notion of stationarity according to which the solutions of (SEL) are the stationary points of F. This notion of stationarity brings coherence to stochastic calculus of variation with respect to stochastic embedding. Finally, we prove a stochastic Noether theorem which introduces an original notion of stochastic first integral. http://front.math.ucdavis.edu/math.PR/0510656 --------------------------------------------------------------- 3919. DENSITY OF PATHS OF ITERATED L\'{E}VY TRANSFORMS OF BROWNIAN MOTION Marc Malric (PMA) The L\'{e}vy transform of a Brownian motion B is the Brownian motion B't, the integral over (O,t) of sign of Bs with respect to dBs. Call T the corresponding transformation on the Wiener space W. We establish that a.s. the orbit of w in W under T is dense in W for the compact uniform convergence topology. http://front.math.ucdavis.edu/math.PR/0511154 --------------------------------------------------------------- 3920. JOINT DENSITY FOR THE LOCAL TIMES OF CONTINUOUS-TIME MARKOV CHAINS D. Brydges and R. van der Hofstad and W. Konig We investigate the local times of a continuous-time Markov chain on an arbitrary discrete state space. For fixed finite range of the Markov chain, we derive an explicit formula for the joint density of all local times on the range, at any fixed time. We use standard tools from the theory of stochastic processes and finite-dimensional complex calculus. We apply this formula in the following directions: (1) we derive large deviation upper estimates for the normalized local times beyond the exponential scale, (2) we derive the upper bound in Varadhan's Lemma for any measurable functional of the local times, (3) we derive large deviation upper bounds for continuous-time simple random walk on large subboxes of $\Z^d$ tending to $\Z^d$ as time diverges, and (4) we prove the analog of the well-known Ray-Knight description of Brownian local times for any nearest-neighbor continuous-time Markov chain on $\Z$, with particularly explicit formulas for simple random walk. http://front.math.ucdavis.edu/math.PR/0511169 --------------------------------------------------------------- 3921. REGENERATIVE REAL TREES Mathilde Weill (DMA) In this work, we give a description of all sigma-finite measures on the space of rooted compact real trees which satisfy a certain regenerative property. We show that any infinite measure which satisfies the regenerative property is the "law" of a Levy tree, that is, the "law" of a tree-valued random variable that describes the genealogy of a population evolving according to a continuous-state branching process. On the other hand, we prove that a probability measure with the regenerative property must be the law of the genealogical tree associated with a continuous-time discrete-state branching process. http://front.math.ucdavis.edu/math.PR/0511172 --------------------------------------------------------------- 3922. PERCOLATION FOR THE STABLE MARRIAGE OF POISSON AND LEBESGUE Marcelo Ventura Freire and Serguei Popov and Marina Vachkovskaia Let $\Xi$ be the set of points (we call the elements of $\Xi$ centers) of Poisson point process in ${\bf R}^d$, $d\geq 2$, with unit intensity. Consider the allocation of ${\bf R}^d$ to $\Xi$ which is stable in the sense of Gale-Shapley marriage problem and in which each center claims a region of volume $\alpha\leq 1$. We prove that there is no percolation in the set of claimed sites if $\alpha$ is small enough, and that, for high dimensions, there is percolation in the set of claimed sites if $\alpha<1$ is large enough. http://front.math.ucdavis.edu/math.PR/0511186 --------------------------------------------------------------- 3923. HARRIS PROCESSES S Sherly and M K Jose and E Sandhya and N Raju In this paper, we develop two stochastic models where the variable under consideration follows Harris distribution. The mean and variance of the processes are derived and the processes are shown to be non- stationary. In the second model, starting with a Poisson process, an alternate way of obtaining Harris process is introduced. http://front.math.ucdavis.edu/math.PR/0510658 --------------------------------------------------------------- 3924. METRIC CONSTRUCTION, STOPPING TIMES AND PATH COUPLING Magnus Bordewich and Martin Dyer and Marek Karpinski In this paper we examine the importance of the choice of metric in path coupling, and the relationship of this to \emph{stopping time analysis}. We give strong evidence that stopping time analysis is no more powerful than standard path coupling. In particular, we prove a stronger theorem for path coupling with stopping times, using a metric which allows us to restrict analysis to standard one-step path coupling. This approach provides insight for the design of non-standard metrics giving improvements in the analysis of specific problems. We give illustrative applications to hypergraph independent sets and SAT instances, hypergraph colourings and colourings of bipartite graphs. http://front.math.ucdavis.edu/math.PR/0511202 --------------------------------------------------------------- 3925. A NECESSARY AND SUFFICIENT CONDITION FOR THE TAIL-TRIVIALITY OF A RECURSIVE TREE PROCESS Antar Bandyopadhyay Given a recursive distributional equation (RDE) and a solution $\mu$ of it, we consider the tree indexed invariant process called the recursive tree process (RTP) with marginal $\mu$. We introduce a new type of bivariate uniqueness property which is different from the one defined by Aldous and Bandyopadhyay (2005), and we prove that this property is equivalent to tail-triviality for the RTP. Thus obtaining a necessary and sufficient condition to determine tail-triviality for a RTP in general. As an application we consider Aldous' (2000) construction of the frozen percolation process on a infinite regular tree and show that the associated RTP has a trivial tail. http://front.math.ucdavis.edu/math.PR/0511203 --------------------------------------------------------------- 3926. HIGH-RESOLUTION PRODUCT QUANTIZATION FOR GAUSSIAN PROCESSES UNDER SUP-NORM DISTORTION Harald Luschgy and Gilles Pag\`{e}s (PMA) We derive high-resolution upper bounds for optimal product quantization of pathwise contionuous Gaussian processes respective to the supremum norm on [0,T]^d. Moreover, we describe a product quantization design which attains this bound. This is achieved under very general assumptions on random series expansions of the process. It turns out that product quantization is asymptotically only slightly worse than optimal functional quantization. The results are applied e.g. to fractional Brownian sheets and the Ornstein-Uhlenbeck process. http://front.math.ucdavis.edu/math.PR/0511208 --------------------------------------------------------------- 3927. INVERSE LITTLEWOOD-OFFORD THEOREMS AND THE CONDITION NUMBER OF RANDOM DISCRETE MATRICES Terence Tao and Van Vu Consider a random sum $\eta_1 v_1 + ... + \eta_n v_n$, where $\eta_1,...,\eta_n$ are i.i.d. random signs and $v_1,...,v_n$ are integers. The Littlewood-Offord problem asks to maximize concentration probabilities such as $\P(\eta_1 v_1 + ... + \eta_n v_n = 0)$ subject to various hypotheses on the $v_1,...,v_n$. In this paper we develop an \emph{inverse} Littlewood- Offord theorem (somewhat in the spirit of Freiman's inverse sumset theorem), which starts with the hypothesis that a concentration probability is large, and concludes that almost all of the $v_1,...,v_n$ are efficiently contained in an arithmetic progression. As an application we give some new bounds on the distribution of the least singular value of a random Bernoulli matrix, which in turn gives upper tail estimates on the condition number. http://front.math.ucdavis.edu/math.PR/0511215 --------------------------------------------------------------- 3928. ON THE SEPARATION PRINCIPLE OF QUANTUM CONTROL Luc Bouten and Ramon van Handel It is well known that continuous quantum measurements and nonlinear filtering can be developed within the framework of the quantum stochastic calculus of Hudson-Parthasarathy. The addition of real-time feedback control has been discussed by many authors, but never in a rigorous way. Here we introduce the notion of a controlled quantum flow, where feedback is taken into account by allowing the coefficients of the quantum stochastic differential equation to be adapted processes in the observation algebra. We then prove a separation theorem for quantum control: the admissible control that minimizes a given cost function is only a function of the filter, provided that the associated Bellman equation has a sufficiently regular solution. Along the way we obtain results on the innovations problem in the quantum setting. http://front.math.ucdavis.edu/math-ph/0511021 --------------------------------------------------------------- 3929. NONLINEARITY, CORRELATION AND THE VALUATION OF EMPLOYEE STOCK OPTIONS M. R. Grasselli We propose a discrete time algorithm for the valuation of employee stock options based on exponential indifference prices and taking into account both the possibility of partial exercise of a fraction of the options and the use of a correlated traded asset to hedge part of their risk. We determine the optimal exercise policy under this conditions and present numerical results showing how both effects can significantly change the value of the option for an employee, as well as its cost for the issuing firm. http://front.math.ucdavis.edu/math.ST/0511234 --------------------------------------------------------------- 3930. AVOIDING DEFEAT IN A BALLS-IN-BINS PROCESS WITH FEEDBACK Roberto Oliveira and Joel Spencer Imagine that there are two bins to which balls are added sequentially, and each incoming ball joins a bin with probability proportional to the p- th power of the number of balls already there. A general result says that if p>1/2, there almost surely is some bin that will have more balls than the other at all large enough times, a property that we call eventual leadership. In this paper, we compute the asymptotics of the probability that bin 1 eventually leads when the total initial number of balls $t$ is large and bin 1 has a fraction \alpha<1/2 of the balls; in fact, this probability is \exp(c_p(\alpha)t + O{t^{2/3}}) for some smooth, strictly negative function c_p. Moreover, we show that conditioned on this unlikely event, the fraction of balls in the first bin can be well-approximated by the solution to a certain ordinary differential equation. http://front.math.ucdavis.edu/math.PR/0510663 --------------------------------------------------------------- 3931. PROBABILITIES ON CLADOGRAMS: INTRODUCTION TO THE ALPHA MODEL Daniel J. Ford The alpha model, a parametrized family of probabilities on cladograms (rooted binary leaf labeled trees), is introduced. This model is Markovian self-similar, deletion-stable (sampling consistent), and passes through the Yule, Uniform and Comb models. An explicit formula is given to calculate the probability of any cladogram or tree shape under the alpha model. Sackin's and Colless' index are shown to be $O(n^{1+\alpha})$ with asymptotic covariance equal to 1. Thus the expected depth of a random leaf with $n$ leaves is $O(n^\alpha)$. The number of cherries on a random alpha tree is shown to be asymptotically normal with known mean and variance. Finally the shape of published phylogenies is examined, using trees from Treebase. http://front.math.ucdavis.edu/math.PR/0511246 --------------------------------------------------------------- 3932. FINITE-DIMENSIONAL APPROXIMATION FOR THE DIFFUSION COEFFICIENT IN SIMPLE EXCLUSION PROCESS M. D. Jara We show that for the mean zero simple exclusion process and for the asymmetric simple exclusion process in dimension d > 2, the self- diffusion coefficient of a tagged particle is stable when approximated by simple exclusion processes on large periodic lattices. The proof relies on a similar property for the Sobolev inner product associated to the generator of the process. http://front.math.ucdavis.edu/math.PR/0511249 --------------------------------------------------------------- 3933. WEAK LOGARITHMIC SOBOLEV INEQUALITIES AND ENTROPIC CONVERGENCE Patrick Cattiaux (MODAL'X and CMAP) and Ivan Gentil (CEREMADE) and Arnaud Guillin (CEREMADE) In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincar\'{e} inequalities, general Beckner inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincar\'{e} inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result. http://front.math.ucdavis.edu/math.PR/0511255 --------------------------------------------------------------- 3934. EXPONENTIAL FUNCTIONALS OF LEVY PROCESSES Jean Bertoin and Marc Yor This text surveys properties and applications of the exponential functional $\int_0^t\exp(-\xi_s)ds$ of real-valued L\'evy processes $\xi=(\xi_t,t \geq0)$. http://front.math.ucdavis.edu/math.PR/0511265 --------------------------------------------------------------- 3935. PROBABILITY & INCOMPRESSIBLE NAVIER-STOKES EQUATIONS: AN OVERVIEW OF SOME RECENT DEVELOPMENTS Edward C. Waymire This is largely an attempt to provide probabilists some orientation to an important class of non-linear partial differential equations in applied mathematics, the incompressible Navier-Stokes equations. Particular focus is given to the probabilistic framework introduced by LeJan and Sznitman [Probab. Theory Related Fields 109 (1997) 343-366] and extended by Bhattacharya et al. [Trans. Amer. Math. Soc. 355 (2003) 5003-5040; IMA Vol. Math. Appl., vol. 140, 2004, in press]. In particular this is an effort to provide some foundational facts about these equations and an overview of some recent results with an indication of some new directions for probabilistic consideration. http://front.math.ucdavis.edu/math.PR/0511266 --------------------------------------------------------------- 3936. SOME RECENT ASPECTS OF RANDOM CONFORMALLY INVARIANT SYSTEMS Wendelin Werner These are the lecture notes from a course given in July 2005 at the summer school in Les Houches. We describe some recent results concerning two-dimensional conformally invariant systems. In particular, we discuss conformally invariant measures on loops and conformal loop-ensembles (CLE). http://front.math.ucdavis.edu/math.PR/0511268 --------------------------------------------------------------- 3937. ON THE ERGODIC PRINCIPLE FOR MARKOV AND QUADRATIC STOCHASTIC PROCESSES AND ITS RELATIONS Nasir Ganikhodjaev and Hasan Akin and Farrukh Mukhamedov In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and only if the associated Markov process satisfies one. http://front.math.ucdavis.edu/math.PR/0511270 --------------------------------------------------------------- 3938. APPROXIMATE MCKEAN-VLASOV REPRESENTATIONS FOR A CLASS OF SPDES Dan Crisan and Jie Xiong The solution $\vartheta =(\vartheta_{t})_{t\geq 0}$ of a class of linear stochastic partial differential equations is approximated using Clark's robust representation approach (\cite{c}, \cite{cc}). The ensuing approximations are shown to coincide with the time marginals of solutions of a certain McKean-Vlasov type equation. We prove existence and uniqueness of the solution of the McKean-Vlasov equation. The result leads to a representation of $\vartheta $as a limit of empirical distributions of systems of equally weighted particles. In particular, the solution of the Zakai equation and that of the Kushner-Stratonovitch equation (the two main equations of nonlinear filtering) are shown to be approximated the empirical distribution of systems of particles that have equal weights (unlike those presented in \cite {kj1} and \cite{kj2}) and do not require additional correction procedures (such as those introduced in \cite{dan3}, \cite{dan4}, \cite{dmm}, etc). http://front.math.ucdavis.edu/math.PR/0510668 --------------------------------------------------------------- 3939. COMPUTABLE CONVERGENCE RATES FOR SUBGEOMETRICALLY ERGODIC MARKOV CHAINS Randal Douc (CMAP) and Eric Moulines (LTCI) and Philippe Soulier (MODAL'X) In this paper, we give quantitative bounds on the $f$-total variation distance from convergence of an Harris recurrent Markov chain on an arbitrary under drift and minorisation conditions implying ergodicity at a sub- geometric rate. These bounds are then specialized to the stochastically monotone case, covering the case where there is no minimal reachable element. The results are illustrated on two examples from queueing theory and Markov Chain Monte Carlo. http://front.math.ucdavis.edu/math.PR/0511273 --------------------------------------------------------------- 3940. ASYMPTOTIC EXPANSION FOR INVERSE MOMENTS OF BINOMIAL AND POISSON DISTRIBUTIONS Marko Znidaric An asymptotic expansion for inverse moments of positive binomial and Poisson distributions is derived. The expansion coefficients of the asymptotic series are given by the positive central moments of the distribution. Compared to previous results, a single expansion formula covers all (also non- integer) inverse moments. In addition, the approach can be generalized to other positive distributions. http://front.math.ucdavis.edu/math.ST/0511226 --------------------------------------------------------------- 3941. EXISTENCE OF THE ZERO RANGE PROCESS AND A DEPOSITION MODEL WITH SUPERLINEAR GROWTH RATES M. Balazs and F. Rassoul-Agha and T. Seppalainen and S. Sethuraman We give a construction of the totally asymmetric zero range process and the so-called bricklayers' process in the attractive case. The novelty is that we allow jump rates to grow as fast as exponentially. These processes have not been constructed for any jump rate growing faster than linearly. We also prove many of the usual semigroup properties, and show that a family of iid. product measures, one for each particle density, is invariant and extremal for the process. Extremality is proved using a new approach, which is rather simple compared to ergodicity proofs found in the literature. http://front.math.ucdavis.edu/math.PR/0511287 --------------------------------------------------------------- 3942. CAPITAL PROCESS AND OPTIMALITY PROPERTIES OF BAYESIAN SKEPTIC IN THE FAIR AND BIASED COIN GAMES Masayuki Kumon and Akimichi Takemura and Kei Takeuchi We study capital process behavior in the fair-coin game and biased- coin games in the framework of the game-theoretic probability of Shafer and Vovk (2001). We show that if Skeptic uses a Bayesian strategy with a beta prior, the capital process is lucidly expressed in terms of the past average of Reality's moves. From this it is proved that the Skeptic's Bayesian strategy weakly forces the strong law of large numbers (SLLN) with the convergence rate of O (\sqrt{\log n/n})$ and if Reality violates SLLN then the exponential growth rate of the capital process is very accurately described in terms of the Kullback divergence between the average of Reality's moves when she violates SLLN and the average when she observes SLLN. We also investigate optimality properties associated with Bayesian strategy. http://front.math.ucdavis.edu/math.ST/0510662 --------------------------------------------------------------- 3943. INTRODUCTION TO DETERMINANTAL POINT PROCESSES FROM A QUANTUM PROBABILITY VIEWPOINT Alex D. Gottlieb Determinantal point processes on a measure space X whose kernels represent trace class Hermitian operators on L^2(X) are associated to "quasifree" density operators on the Fock space over L^2(X). http://front.math.ucdavis.edu/math.PR/0511334 --------------------------------------------------------------- 3944. A NOTE ON A.S. FINITENESS OF PERPETUAL INTEGRAL FUNCTIONALS OF DIFFUSIONS Paavo Salminen and Marc Yor (PMA) In this note, with the help of the boundary classification of diffusions, we derive a criterion of the convergence of perpetual integral functionals of transient real-valued diffusions. In the particular case of transient Bessel processes, we note that this criterion agrees with the one obtained via Jeulin's convergence lemma. http://front.math.ucdavis.edu/math.PR/0511336 --------------------------------------------------------------- 3945. SEQUENTIAL AND ASYNCHRONOUS PROCESSES DRIVEN BY STOCHASTIC OR QUANTUM GRAMMARS AND THEIR APPLICATION TO GENOMICS: A SURVEY Dimitri Petritis (IRMAR) We present the formalism of sequential and asynchronous processes defined in terms of random or quantum grammars and argue that these processes have relevance in genomics. To make the article accessible to the non-mathematicians, we keep the mathematical exposition as elementary as possible, focusing on some general ideas behind the formalism and stating the implications of the known mathematical results. We close with a set of open challenging problems. http://front.math.ucdavis.edu/math.PR/0511346 --------------------------------------------------------------- 3946. NON-TANGENTIAL AND PROBABILISTIC BOUNDARY BEHAVIOR OF PLURIHARMONIC FUNCTIONS Steve Tanner Let $u$ be a pluriharmonic function on the unit ball in $C^n$. I consider the relationship between the set of points $L_u$ on the boundary of the ball at which $u$ converges non-tangentially, and the set of points $\L_u$ at which $u$ converges along conditioned Brownian paths. For harmonic funcitons $u $ of two variables, the result $L_u = \L_u$ (a.e.) has been known for some time, as has a counterexample to the same equality for three variable harmonic functions. I extend the $L_u = \L_u$ (a.e.) result to pluriharmonic functions in arbitrary dimensions. http://front.math.ucdavis.edu/math.PR/0511368 --------------------------------------------------------------- 3947. SHARP ASYMPTOTIC BEHAVIOR FOR WETTING MODELS IN (1+1)-DIMENSION Francesco Caravenna and Giambattista Giacomin and Lorenzo Zambotti We consider continuous and discrete (1+1)-dimensional wetting models which undergo a localization/delocalization phase transition. Using a simple approach based on Renewal Theory we determine the precise asymptotic behavior of the partition function, from which we obtain the scaling limits of the models and an explicit construction of the infinite volume measure (thermodynamic limit) in all regimes, including the critical one. http://front.math.ucdavis.edu/math.PR/0511376 --------------------------------------------------------------- 3948. LACE EXPANSION FOR THE ISING MODEL Akira Sakai The lace expansion has been a powerful tool to investigate mean-field behavior for various stochastic-geometrical models, such as self- avoiding walk and percolation, above their respective upper-critical dimension. In this paper, we prove for the first time the lace expansion for the Ising model, which is independent of the property of the spin-spin coupling. In the ferromagnetic case, we provide key propositions to prove that, without requiring the reflection positivity of the spin-spin coupling, the two-point function obeys a Gaussian infrared bound for the nearest-neighbor model with d>>4 and for the spread-out model with d>4 and L>>1, as well as that the critical two-point function exhibits a Gaussian asymptotics for the spread-out model with d>4 and L>>1. As a result, these models exhibit the ferromagnetic mean-field behavior. http://front.math.ucdavis.edu/math-ph/0510093 --------------------------------------------------------------- 3949. ASYMPTOTICS OF COUNTS OF SMALL COMPONENTS IN RANDOM COMBINATORIAL STRUCTURES AND MODELS OF COAGULATION-FRAGMENTATION Boris L. Granovsky We establish necessary and sufficient conditions for convergence of non scaled multiplicative measures on the set of partitions. The measures depict component spectrums of random structures and the equilibrium of some models of statistical mechanics, including stochastic processes of coagulation-fragmentation. Based on the above result, we show that the common belief that interacting groups in mean field models become independent as the number of particles goes to infinity, is in general not true. http://front.math.ucdavis.edu/math.PR/0511381 --------------------------------------------------------------- 3950. REMARKS ON SOME LINEAR FRACTIONAL STOCHASTIC EQUATIONS Ivan Nourdin (PMA) and Ciprian A. Tudor (SAMOS) Using the multiple stochastic integrals we prove an existence and uniqueness result for a linear stochastic equation driven by the fractional Brownian motion with any Hurst parameter. We study both the one parameter and two parameter cases. When the drift is zero, we show that in the one- parameter case the solution in an exponential, thus positive, function while in the two-parameter settings the solution is negative on a non-negligible set. http://front.math.ucdavis.edu/math.PR/0511383 --------------------------------------------------------------- 3951. FRAGMENTATION OF COMPOSITIONS AND INTERVALS Anne-Laure Basdevant (PMA) The fragmentation processes of exchangeable partitions have already been studied by several authors. In this paper, we examine rather fragmentation of exchangeable compositions, that means partitions of $\mcn$ where the order of the blocks counts. We will prove that such a fragmentation is bijectively associated to an interval fragmentation. Using this correspondence, we then calculate the Hausdorff dimension of certain random closed set that arise in interval fragmentations and we study Ruelle's interval fragmentation. http://front.math.ucdavis.edu/math.PR/0511388 --------------------------------------------------------------- 3952. AN INVARIANCE PRINCIPLE FOR AZ\'{E}MA MARTINGALES Nathanael Enriquez (PMA) An invariance principle for Az\'{e}ma martingales is presented as well as a new device to construct solutions of Emery's structure equations. http://front.math.ucdavis.edu/math.PR/0511402 --------------------------------------------------------------- 3953. THEORY OF AMALGAMATED LP SPACES IN NONCOMMUTATIVE PROBABILITY Marius Junge and Javier Parcet Let $f_1, f_2, ..., f_n$ be a family of independent copies of a given random variable $f$ in a probability space $(\Omega, \mathcal{F}, \mu)$. Then, the following equivalence of norms holds whenever $1 \le q \le p < \infty$ $$\Big(\int_{\Omega} \Big[ \sum_{k=1}^n |f_k|^q \Big]^{\frac{p}{q}} d \mu \Big)^{\frac1p} \sim \max_{r \in \{p,q\}} {n^{\frac1r} \Big(\int_ \Omega |f|^r d\mu \Big)^{\frac1r}}.$$ We prove a noncommutative analogue of this inequality for sums of free random variables over a given von Neumann subalgebra. This formulation leads to new classes of noncommutative function spaces which appear in quantum probability as square functions, conditioned square functions and maximal functions. Our main tools are Rosenthal type inequalities for free random variables, noncommutative martingale theory and factorization of operator-valued analytic functions. This allows us to generalize this inequality as a result for noncommutative $L_p$ in the category of operator spaces. Moreover, the use of free random variables produces the right formulation for $p=\infty$, which has not a commutative counterpart. http://front.math.ucdavis.edu/math.OA/0511406 From pas at www.economia.unimi.it Wed Mar 1 10:18:34 2006 From: pas at www.economia.unimi.it (pas@www.economia.unimi.it) Date: Wed Mar 1 11:18:58 2006 Subject: [Pas] Probability Abstracts 91 Message-ID: <570F3E40-7A66-4C70-B0DE-B4ABBDDABF0C@unimi.it> March 1, 2006 Letter 91 Probability Abstract Service Abstracts from Jan-1-2006 to Feb-28-2006 --------------------------------------------------------------- 3954. FLUID LIMIT OF A HEAVILY LOADED EDF QUEUE WITH IMPATIENT CUSTOMERS Laurent Decreusefond and Pascal Moyal In this paper we present the fluid limit of an heavily loaded Earliest Deadline First queue with impatient customers, represented by a measure-valued process keeping track of residual time-credits of lost and waiting customers. This fluid limit is the solution of an integrated transport equation. We then use this fluid limit to derive fluid approximations of the processes counting the number of waiting and already lost customers. http://front.math.ucdavis.edu/math.PR/0512660 --------------------------------------------------------------- 3955. ANALYSIS OF DISK SCHEDULING, INCREASING SUBSEQUENCES AND SPACE- TIME GEOMETRY Eitan Bachmat We consider the problem of estimating the average tour length of the asymmetric TSP arising from the disk scheduling problem with a linear seek function and a probability distribution on the location of I/O requests. The optimal disk scheduling algorithm of Andrews, Bender and Zhang is interpreted as a simple peeling process on points in a 2 dimensional space-time w.r.t the causal structure. The patience sorting algorithm for finding the longest increasing subsequence in a permutation can be given a similar interpretation. Using this interpretation we show that the optimal tour length is the length of the maximal curve with respect to a Lorentzian metric on the surface of the disk drive. This length can be computed explicitly in some interesting cases. When the probability distribution is assumed uniform we provide finer asymptotics for the tour length. The interpretation also provides a better understanding of patience sorting and allows us to extend a result of Aldous and Diaconis on pile sizes http://front.math.ucdavis.edu/math.OC/0601025 --------------------------------------------------------------- 3956. ASYMPTOTICS OF BERNOULLI RANDOM WALKS, BRIDGES, EXCURSIONS AND MEANDERS WITH A GIVEN NUMBER OF PEAKS Jean-Maxime Labarbe (LM-Versailles) and Jean-Fran\c{c}ois Marckert (LaBRI) A Bernoulli random walk is a random trajectory starting from 0 and having i.i.d. increments, each of them being $+1$ or -1, equally likely. The other families cited in the title are Bernoulli random walks under various conditionings. A peak in a trajectory is a local maximum. In this paper, we condition the families of trajectories to have a given number of peaks. We show that, asymptotically, the main effect of setting the number of peaks is to change the order of magnitude of the trajectories. The counting process of the peaks, that encodes the repartition of the peaks in the trajectories, is also studied. It is shown that suitably normalized, it converges to a Brownian bridge which is independent of the limiting trajectory. Applications in terms of plane trees and parallelogram polyominoes are also provided. http://front.math.ucdavis.edu/math.PR/0601624 --------------------------------------------------------------- 3957. THERMAL CONDUCTIVITY FOR A MOMENTUM CONSERVING MODEL Giada Basile (CEREMADE) and Cedric Bernardin (UMPA-ENSL) and Stefano Olla (CEREMADE) We present here complete mathematical proofs of the results announced in cond-mat/0509688. We introduce a model whose thermal conductivity diverges in dimension 1 and 2, while it remains finite in dimension 3. We consider a system of harmonic oscillators perturbed by a stochastic dynamics conserving momentum and energy. We compute the nite-size thermal conductivity via Green-Kubo formula. In the limit as the size N of the system goes to in nity, conductivity diverges like N in dimension 1 and like lnN in dimension 2. Conductivity remains finite if dimesion is 3 or higher or if a pinning (on site potential) is present. http://front.math.ucdavis.edu/cond-mat/0601544 --------------------------------------------------------------- 3958. STRUCTURE THEOREM FOR (D,G,H)-MAPS Alex V. Kontorovich and Yakov G. Sinai The (3x+1)-Map, T, acts on the set, Pi, of positive integers not divisible by 2 or 3. It is defined by T(x) = (3x+1)/2^k, where k is the largest integer for which T(x) is an integer. The (3x+1)-Conjecture asks if for every x in Pi there exists an integer, n, such that T^n (x) = 1. The Statistical (3x+1)- Conjecture asks the same question, except for a subset of Pi of density 1. The Structure Theorem proven in \cite{sinai} shows that infinity is in a sense a repelling point, giving some reasons to expect that the (3x+1)-Conjecture may be true. In this paper, we present the analogous theorem for some generalizations of the (3x+1)-Map, and expand on the consequences derived in \cite{sinai}. The generalizations we consider are determined by positive coprime integers, d and g, with g > d >= 2, and a periodic function, h(x). The map T is defined by the formula T(x) = (gx+h(gx))/d^k, where k is again the largest integer for which T(x) is an integer. We prove an analogous Structure Theorem for (d,g,h)-Maps, and that the probability distribution corresponding to the density converges to the Wiener measure with the drift log(g) - d/(d-1)log(d) and positive diffusion constant. This shows that it is natural to expect that typical trajectores return to the origin if log(g) - d/(d-1) log(d) <0 and escape to infinity otherwise. http://front.math.ucdavis.edu/math.NT/0601622 --------------------------------------------------------------- 3959. CAPITAL REQUIREMENT FOR ACHIEVING ACCEPTABILITY Soumik Pal Consider an agent who enters a financial market on day t = 0 with an initial capital amount x. He invests this amount on stocks and the money market, and by day t = T, has generated a wealth W . He is given a convex class of probability measures (called scenarios) and a real-valued function (or floors) corresponding to each scenario. The agent faces the constraints that the expectation of W under each scenario must not be less than the corresponding floor. We call x acceptable if one can start with x and successfully generate W satisfying these constraints. The set of acceptable x is a half-line in R, unbounded from above. We show that under some regularity conditions on the set of scenarios and the floor function, the infimum of this set is given by the supremum of the floors over all scenarios under which S is a martingale. http://front.math.ucdavis.edu/math.PR/0601627 --------------------------------------------------------------- 3960. DIFFERENTIAL EQUATIONS DRIVEN BY H\"{O}LDER CONTINUOUS FUNCTIONS OF ORDER GREATER THAN 1/2 Yaozhong Hu and David Nualart We derive estimates for the solutions to differential equations driven by a H\"older continuous function of order $\beta>1/2$. As an application we deduce the existence of moments for the solutions to stochastic partial differential equations driven by a fractional Brownian motion with Hurst parameter $H>{1/2}$. http://front.math.ucdavis.edu/math.PR/0601628 --------------------------------------------------------------- 3961. CONFIGURATIONS OF BALLS IN EUCLIDEAN SPACE THAT BROWNIAN MOTION CANNOT AVOID Tom Carroll and Joaquim Ortega-Cerd\`a We consider a collection of balls in Euclidean space and the problem of determining if Brownian motion has a positive probability of avoiding all the balls http://front.math.ucdavis.edu/math.PR/0601632 --------------------------------------------------------------- 3962. DISCRETE LOGISTIC BRANCHING POPULATIONS AND THE CANONICAL DIFFUSION OF ADAPTIVE DYNAMICS Nicolas Champagnat (WIAS) and Amaury Lambert (FESE) The biological theory of adaptive dynamics proposes a description of the long-time evolution of an asexual population, based on the assumptions of large population, rare mutations and small mutation steps, that lead to a deterministic ODE, called 'canonical equation of adaptive dynamics'. However, in order to include the effect of genetic drift in this description, we have to apply a limit of weak selection to a finite stochastically fluctuating discrete population subject to competition in the logistic branching fashion. We start with the study of the particular case of two competing subpopulations resident and mutant) and seek explicit first-order formulae for the probability of fixation of the mutant, also interpreted as the mutant's fitness, in the vicinity of neutrality. In particular, the first-order term is a linear combination of products of functions of the initial mutant frequency times functions of the initial total population size, called invasibility coefficients (fertility, defence, aggressiveness, isolation, survival). Then we apply a limit of rare mutations to a population subject to mutation, birth and competition where the number of coexisting types may fluctuate, while keeping the population size finite. This leads to a jump process, the so- called 'trait substitution sequence', where evolution proceeds by successive invasions and fixations of mutant types. Finally, we apply a limit of weak selection (small mutation steps) to this jump process, that leads to a diffusion process of evolution, called 'canonical diffusion of adaptive dynamics', in which genetic drift is combined with directional selection driven by the fitness gradient. http://front.math.ucdavis.edu/math.PR/0601643 --------------------------------------------------------------- 3963. THE MAXIMUM ENTROPY ANSATZ IN THE ABSENCE OF A TIME ARROW: FRACTIONAL POLE MODELS Tryphon T. Georgiou The maximum entropy ansatz, as it is often invoked in the context of time-series analysis, suggests the selection of a power spectrum which is consistent with autocorrelation data and corresponds to a random process least predictable from past observations. We introduce and compare a class of spectra with the property that the underlying random process is least predictable at any given point from the complete set of past and future observations. In this context, randomness is quantified by the size of the corresponding smoothing error and deterministic processes are characterized by integrability of the inverse of their power spectral densities--as opposed to the log- integrability in the classical setting. The power spectrum which is consistent with a partial autocorrelation sequence and corresponds to the most random process in this new sense, is no longer rational but generated by finitely many fractional-poles. http://front.math.ucdavis.edu/math.PR/0601648 --------------------------------------------------------------- 3964. SYMMETRIZATION OF BERNOULLI Soumik Pal Let X be a random variable. We shall call an independent random variable Y to be a symmetrizer for X, if X+Y is symmetric around zero. A random variable is said to be symmetry resistant if the variance of any symmetrizer Y, is never smaller than the variance of X itself. We prove that a Bernoulli(p) random variable is symmetry resistant if and only if p is not 1/2. This is an old problem proved in 1999 by Kagan, Mallows, Shepp, Vanderbei & Vardi using linear programming principles. We reprove it here using completely probabilistic tools using Skorokhod embedding and Ito's rule. http://front.math.ucdavis.edu/math.PR/0601652 --------------------------------------------------------------- 3965. ON THE LIMITING VELOCITY OF HIGH-DIMENTIONAL RANDOM WALK IN RANDOM ENVIRONMENT Noam Berger We show that Random Walk in uniformly elliptic i.i.d. environment in dimension 5 and higher has at most one non-zero limiting velocity. In particular this proves a law of large numbers in the distributionally symmetric case and establishes connections between different conjectures. http://front.math.ucdavis.edu/math.PR/0601656 --------------------------------------------------------------- 3966. SMALL-TIME BEHAVIOR OF BETA COALESCENTS Julien Berestycki and Nathanael Berestycki and Jason Schweinsberg For a finite measure $\Lambda$ on $[0,1]$, the $\Lambda$-coalescent is a coalescent process such that, whenever there are $b$ clusters, each $k $-tuple of clusters merges into one at rate $\int_0^1 x^{k-2} (1-x)^{b-k} \Lambda(dx)$. It has recently been shown that if $1 < \alpha < 2$, the $\Lambda$- coalescent in which $\Lambda$ is the Beta$(2-\alpha, \alpha)$ distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an $\alpha$-stable branching mechanism. Here we use facts about CSBPs to establish new results about the small-time asymptotics of beta coalescents. We prove an a.s. limit theorem for the number of blocks at small times, and we establish results about the sizes of the blocks. We also calculate the Hausdorff and packing dimensions of a metric space associated with the beta coalescents, and we find the sum of the lengths of the branches in the coalescent tree, both of which are determined by the behavior of coalescents at small times. We extend most of these results to other $\Lambda$-coalescents for which $ \Lambda$ has the same asymptotic behavior near zero as the Beta$(2-\alpha, \alpha)$ distribution. This work complements recent work of Bertoin and Le Gall, who also used CSBPs to study small-time properties of $\Lambda$-coalescents. http://front.math.ucdavis.edu/math.PR/0601032 --------------------------------------------------------------- 3967. REFLECTING A LANGEVIN PROCESS AT AN ABSORBING BOUNDARY Jean Bertoin (PMA) We consider a Langevin process with white noise random forcing. We suppose that the energy of the particle is instantaneously absorbed when it hits some fixed obstacle. We show that nonetheless, the particle can be instantaneously reflected, and study some properties of this reflecting solution. http://front.math.ucdavis.edu/math.PR/0601657 --------------------------------------------------------------- 3968. STRONG DISORDER IMPLIES STRONG LOCALIZATION FOR DIRECTED POLYMERS IN A RANDOM ENVIRONMENT Philippe Carmona (LMJL) and Yueyun Hu (LAGA) In this note we show that in any dimension $d$, the strong disorder property implies the strong localization property. This is established for a continuous time model of directed polymers in a random environment : the parabolic Anderson Model. http://front.math.ucdavis.edu/math.PR/0601670 --------------------------------------------------------------- 3969. BROWNIAN LOCAL MINIMA, RANDOM DENSE COUNTABLE SETS AND RANDOM EQUIVALENCE CLASSES Boris Tsirelson A random dense countable set is characterized (in distribution) by independence and stationarity. Two examples are `Brownian local minima' and `unordered infinite sample'. They are identically distributed. A framework for such concepts, proposed here, includes a wide class of random equivalence classes. http://front.math.ucdavis.edu/math.PR/0601673 --------------------------------------------------------------- 3970. POSITIONAL GAMES ON RANDOM GRAPHS Milos Stojakovic and Tibor Szabo We introduce and study Maker/Breaker-type positional games on random graphs. Our main concern is to determine the threshold probability $p_{F}$ for the existence of Maker's strategy to claim a member of $F$ in the unbiased game played on the edges of random graph $G(n,p)$, for various target families $F$ of winning sets. More generally, for each probability above this threshold we study the smallest bias $b$ such that Maker wins the $(1\:b)$ biased game. We investigate these functions for a number of basic games, like the connectivity game, the perfect matching game, the clique game and the Hamiltonian cycle game. http://front.math.ucdavis.edu/math.CO/0601659 --------------------------------------------------------------- 3971. HADAMARD FUNCTIONS OF INVERSE M-MATRICES Claude Dellacherie and Servet Martinez and Jaime San Martin We prove that the class of GUM matrices is the largest class of bi- potential matrices stable under Hadamard increasing functions. We also show that any power greater than 1, in the sense of Hadamard functions, of an inverse M-matrix is also inverse M-matrix showing a conjecture stated in Neumann 1998. We study the class of filtered matrices, which include naturally the GUM matrices, and present some sufficient conditions for a filtered matrix to be a bi-potential. http://front.math.ucdavis.edu/math.PR/0601688 --------------------------------------------------------------- 3972. MULTI-DIMENSIONAL G-BROWNIAN MOTION AND RELATED STOCHASTIC CALCULUS UNDER G-EXPECTATION Shige Peng We develop a notion of nonlinear expectation --G-expectation-- generated by a nonlinear heat equation with infinitesimal generator G. We first study multi-dimensional G-normal distributions. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a multi dimensional G-Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Ito's type with respect to our G-Brownian motion and derive the related Ito's formula. We have also obtained the existence and uniqueness of stochastic differential equation under our G-expectation. http://front.math.ucdavis.edu/math.PR/0601699 --------------------------------------------------------------- 3973. AN INTRODUCTION TO QUANTUM FILTERING Luc Bouten and Ramon van Handel and Matthew James This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as a least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. We describe the quantum It\^o calculus and its use in the modelling of physical systems. We use both reference probability and innovations methods to obtain quantum filtering equations for system-probe models from quantum optics. http://front.math.ucdavis.edu/math.OC/0601741 --------------------------------------------------------------- 3974. $G$--EXPECTATION, $G$--BROWNIAN MOTION AND RELATED STOCHASTIC CALCULUS OF IT\^{O}'S TYPE Shige Peng We introduce a notion of nonlinear expectation --$G$--expectation-- generated by a nonlinear heat equation with infinitesimal generator $G$. We first discuss the notion of $G$--standard normal distribution. With this nonlinear distribution we can introduce our $G$--expectation under which the canonical process is a $G$--Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of It\^{o}'s type with respect to our $G$--Brownian motion and derive the related It\^{o}'s formula. We have also give the existence and uniqueness of stochastic differential equation under our $G$--expectation. As compared with our previous framework of $g$-- expectations, the theory of $G$--expectation is intrinsic in the sense that it is not based on a given (linear) probability space. http://front.math.ucdavis.edu/math.PR/0601035 --------------------------------------------------------------- 3975. METASTABLE BEHAVIOUR OF SMALL NOISE LEVY-DRIVEN DIFFUSION Ilya Pavlyukevich We consider a dynamical system in R driven by a vector field -U', where U is a multi-well potential satisfying some regularity conditions. We perturb this dynamical system by a Levy noise of small intensity and such that the heaviest tail of its Levy measure is regularly varying. We show that the perturbed dynamical system exhibits metastable behaviour i.e. on a proper time scale it reminds of a Markov jump process taking values in the local minima of the potential U. Due to the heavy-tail nature of the random perturbation, the results differ strongly from the well studied purely Gaussian case. http://front.math.ucdavis.edu/math.PR/0601771 --------------------------------------------------------------- 3976. FUNCTIONAL QUANTIZATION RATE AND MEAN PATHWISE REGULARITY OF PROCESSES WITH AN APPLICATION TO L\'{E}VY PROCESSES Harald Luschgy (PMA) and Gilles Pag\`{e}s (PMA) We investigate the connections between the mean pathwise regularity of stochastic processes and their $L^r(\P$)-functional quantization rate as random variables taking values in some $L^p([0,T],dt)$-spaces ($<0p\le r$). Our main tool is the Haar basis. We then emphasize that the derived functional quantization rate may be optimal (like for the Brownian motion) or not (like for the Poisson process). Then, we focus on the specific family of L \'evy processes for which we derive a general quantization rate based on the regular variation properties of its L\'evy measure at 0. The case of compound Poisson processes which appears as degenerate in the former approach, are studied specifically: one observes some rates which are in-between finite dimensional and infinite dimensional "usual" rates. http://front.math.ucdavis.edu/math.PR/0601774 --------------------------------------------------------------- 3977. PERPETUAL INTEGRAL FUNCTIONALS OF DIFFUSIONS AND THEIR NUMERICAL COMPUTATIONS P. Salminen and O. Wallin In this paper we study perpetual integral functionals of diffusions. Our interest is focused on cases where such functionals can be expressed as first hitting times for some other diffusions. In particular, we generalize the result which connects one-sided functionals of Brownian motion with drift with first hitting times of reflecting diffusions. Interpretating perpetual integral functionals as hitting times allows us to compute numerically their distributions by applying numerical algorithms for hitting times. Hereby, we discuss two approaches: the numerical inversion of the Laplace transform of the first hitting time and the numerical solution of the PDE associated with the distribution function of the first hitting time. For numerical inversion of Laplace tranforms we have implemented the Euler algorithm developed by Abate and Whitt. However, perpetuities lead often to diffusions for which the explicit forms of the Laplace transforms of first hitting times are not available. In such cases, and also otherwise, algorithms for numerical solutions of PDE's can be evoked. In particular, we analyze the Kolmogorov PDE of some diffusions appearing in our work via the Crank- Nicolson scheme. http://front.math.ucdavis.edu/math.PR/0601775 --------------------------------------------------------------- 3978. PROPAGATION OF MEMORY PARAMETER FROM DURATIONS TO COUNTS Rohit Deo (IOMS) and Clifford M. Hurvich (IOMS) and Philippe Soulier (MODAL'X), Yi Wang (IOMS) We establish sufficient conditions on durations that are stationary with finite variance and memory parameter $d \in [0,1/2)$ to ensure that the corresponding counting process $N(t)$ satisfies $\textmd{Var} N(t) \sim C t^{2d+1}$ ($C>0$) as $t \to \infty$, with the same memory parameter $d \in [0,1/2)$ that was assumed for the durations. Thus, these conditions ensure that the memory in durations propagates to the same memory parameter in counts and therefore in realized volatility. We then show that any utoregressive Conditional Duration ACD(1,1) model with a sufficient number of finite moments yields short memory in counts, while any Long Memory Stochastic Duration model with $d>0$ and all finite moments yields long memory in counts, with the same $d$. http://front.math.ucdavis.edu/math.ST/0601742 --------------------------------------------------------------- 3979. MODERATE DEVIATIONS FOR THE RANGE OF PLANAR RANDOM WALKS Richard F. Bass and Xia Chen and and Jay Rosen Given a symmetric random walk in $Z^2$ with finite second moments, let $R_n$ be the range of the random walk up to time $n$. We study moderate deviations for $R_n -E R_n$ and $E R_n -R_n$. We also derive the corresponding laws of the iterated logarithm. http://front.math.ucdavis.edu/math.PR/0602001 --------------------------------------------------------------- 3980. THE BROWNIAN FRAME PROCESS AS A ROUGH PATH Benjamin Hoff We introduce the (path-valued) Brownian frame process whose evaluation at time t is the sample path of the underlying Brownian motion run from time t-1 to t. Due to its connections with Gaussian Volterra processes and SDDEs this is an interesting object to study. The first part deals with path-wise properties of the Brownian frame process in the p-variation norm. The second part shows the non-existence of a Levy area random variable in a particular norm, revealing the difficulty in establishing a Rough Path integration theory for the Brownian Frame process. http://front.math.ucdavis.edu/math.PR/0602008 --------------------------------------------------------------- 3981. A DATA-RECONSTRUCTED FRACTIONAL VOLATILITY MODEL Rui Vilela Mendes Based on criteria of mathematical simplicity and consistency with empirical market data, a stochastic volatility model is constructed, the volatility process being driven by fractional noise. Price return statistics and asymptotic behavior are derived from the model and compared with data. http://front.math.ucdavis.edu/math.PR/0602013 --------------------------------------------------------------- 3982. BOUNDS ON REGENERATION TIMES AND LIMIT THEOREMS FOR SUBGEOMETRIC MARKOV CHAINS Randal Douc (CMAP) and Arnaud Guillin (CEREMADE) and Eric Moulines (LTCI) This paper studies limit theorems for Markov Chains with general state space under conditions which imply subgeometric ergodicity. We obtain a central limit theorem and moderate deviation principles for additive not necessarily bounded functional of the Markov chains under drift and minorization conditions which are weaker than the Foster-Lyapunov conditions. The regeneration- split chain method and a precise control of the modulated moment of the hitting time to small sets are employed in the proof. http://front.math.ucdavis.edu/math.PR/0601036 --------------------------------------------------------------- 3983. EXACT CONDITIONS FOR COUNTABLE INCLUSION-EXCLUSION IDENTITY AND EXTENSIONS Shmuel Friedland and Elliot Krop We give simple necessary and sufficient conditions for the inclusion-exclusion identity to hold for an infinite countable number of sets. In terms of a random variable, whose range are nonnegative integers, this condition is equivalent to the convergence to zero of binomial moments. Some standard extensions of the countable inclusion-exclusion identity are also given. http://front.math.ucdavis.edu/math.PR/0602035 --------------------------------------------------------------- 3984. ON THE SPEED OF THE ONE-DIMENSIONAL EXCITED RANDOM WALK IN THE TRANSIENT REGIME Thomas Mountford and Leandro P. R. Pimentel and Glauco Valle We study a class of nearest-neighbor discrete time integer random walks introduced by Zerner, the so called multi-excited random walks. The jump probabilities for such random walker have a drift to the right whose intensity depends on a random or non-random environment that also evolves in time according to the last visited site. A complete description of the recurrence and transience phases was given by Zerner under fairly general assumptions for the environment. We contribute in this paper with some results that allows us to point out if the random walker speed is strictly positive or not in the transient case for a class of non-random environments. http://front.math.ucdavis.edu/math.PR/0602041 --------------------------------------------------------------- 3985. ROUGH PATH ANALYSIS VIA FRACTIONAL CALCULUS Yaozhong Hu and David Nualart Using fractional calculus we define integrals of the form $% \int_{a}^{b}f(x_{t})dy_{t}$, where $x$ and $y$ are vector-valued H \"{o}lder continuous functions of order $\displaystyle \beta \in (\frac13, \frac12)$ and $f$ is a continuously differentiable function such that $f'$ is $\lambda$-H\"oldr continuous for some $\lambda>\frac1\beta-2$. Under some further smooth conditions on $f$ the integral is a continuous functional of $x$, $y$, and the tensor product $x\otimes y$ with respect to the H \"{o}lder norms. We derive some estimates for these integrals and we solve differential equations driven by the function $y$. We discuss some applications to stochastic integrals and stochastic differential equations. http://front.math.ucdavis.edu/math.PR/0602050 --------------------------------------------------------------- 3986. VARIATIONAL BOUNDS FOR THE GENERALIZED RANDOM ENERGY MODEL Cristian Giardina' and Shannon Starr We compute the pressure of the random energy model (REM) and generalized random energy model(GREM) by establishing variational upper and lower bounds. For the upper bound, we generalize Guerra's ``broken replica symmetry bounds",and identify the random probability cascade as the appropriate random overlap structure for the model. For the REM the lower bound is obtained, in the high temperature regime using Talagrand's concentration of measure inequality, and in the low temperature regime using convexity and the high temperature formula. The lower bound for the GREM follows from the lower bound for the REM by induction. While the argument for the lower bound is fairly standard, our proof of the upper bound is new. http://front.math.ucdavis.edu/math-ph/0601068 --------------------------------------------------------------- 3987. A CORRESPONDENCE PRINCIPLE BETWEEN (HYPER)GRAPH THEORY AND PROBABILITY THEORY, AND THE (HYPER)GRAPH REMOVAL LEMMA Terence Tao We introduce a correspondence principle (analogous to the Furstenberg correspondence principle) that allows one to extract an infinite random graph or hypergraph from a sequence of increasingly large deterministic graphs or hypergraphs. As an application we present a new (infinitary) proof of the hypergraph removal lemma of Nagle-Schacht-R\"odl-Skokan and Gowers, which does not require the hypergraph regularity lemma and requires significantly less computation. This in turn gives new proofs of several corollaries of the hypergraph removal lemma, such as Szemer\'edi's theorem on arithmetic progressions. http://front.math.ucdavis.edu/math.CO/0602037 --------------------------------------------------------------- 3988. PATHWISE STATIONARY SOLUTIONS OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS AND BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS ON INFINITE HORIZON Qi Zhang and Huaizhong Zhao The main purpose of this paper is to study the existence of stationary solution for stochastic partial differential equations. We establish a new connection between backward doubly stochastic differential equations on infinite time horizon and the stationary solution of the SPDEs. For this we study the existence of the solution of the associated BDSDEs on infinite time horizon and prove it is a stationary viscosity solution of the corresponding SPDEs. http://front.math.ucdavis.edu/math.PR/0602054 --------------------------------------------------------------- 3989. ON THE DECAY OF FRAGMENTS IN HOMOGENEOUS FRAGMENTATIONS Nathalie Krell (PMA) We consider a mass-conservative fragmentation of the unit interval. The main purpose of this work is to specify the Hausdorff dimension of the set of locations having exactly an exponential decay. The study relies on an additive martingale which arises naturally in this setting, and a class of L \'{e}vy process constrained to stay in a finite interval. http://front.math.ucdavis.edu/math.PR/0602065 --------------------------------------------------------------- 3990. LARGE DEVIATIONS ESTIMATES FOR SELF-INTERSECTION LOCAL TIMES FOR SIMPLE RANDOM WALK IN $\Z^3$ Amine Asselah We obtain large deviations estimates for the self-intersection local times for a symmetric random walk in dimension 3. Also, we show that the main contribution to making the self-intersection large, in a time period of length $n$, comes from sites visited less than some power of $\log(n)$. This is opposite to the situation in dimensions larger or equal to 5. Finally, we present two applications of our estimates: (i) to moderate deviations estimates for the range of a random walk, and (ii) to moderate deviations for random walk in random sceneries. http://front.math.ucdavis.edu/math.PR/0602074 --------------------------------------------------------------- 3991. EXACT RATE OF CONVERGENCE OF SOME APPROXIMATION SCHEMES ASSOCIATED TO SDES DRIVEN BY A FRACTIONAL BROWNIAN MOTION Andreas Neuenkirch (TU DARMSTADT) and Ivan Nourdin (PMA) In this paper, we derive the exact rate of convergence of some approximation schemes associated to scalar stochastic differential equations driven by a fractional Brownian motion with Hurst index H. http://front.math.ucdavis.edu/math.PR/0601038 --------------------------------------------------------------- 3992. MOMENTS OF CONVEX DISTRIBUTION FUNCTIONS AND COMPLETELY ALTERNATING SEQUENCES Alexander Gnedin and Jim Pitman We solve the moment problem for convex distribution functions on $[0,1]$ in terms of completely alternating sequences. This complements a recent solution of this problem by Diaconis and Freedman, and relates this work to the L{\'e}vy-Khintchine formula for the Laplace transform of a subordinator, and to regenerative composition structures. http://front.math.ucdavis.edu/math.PR/0602091 --------------------------------------------------------------- 3993. EXCLUSION PROCESSES IN HIGHER DIMENSIONS: STATIONARY MEASURES AND CONVERGENCE M. Bramson and T. M. Liggett There has been significant progress recently in our understanding of the stationary measures of the exclusion process on $Z$. The corresponding situation in higher dimensions remains largely a mystery. In this paper we give necessary and sufficient conditions for a product measure to be stationary for the exclusion process on an arbitrary set, and apply this result to find examples on $Z^d$ and on homogeneous trees in which product measures are stationary even when they are neither homogeneous nor reversible. We then begin the task of narrowing down the possibilities for existence of other stationary measures for the process on $Z^d$. In particular, we study stationary measures that are invariant under translations in all directions orthogonal to a fixed nonzero vector. We then prove a number of convergence results as $t\to \infty$ for the measure of the exclusion process. Under appropriate initial conditions, we show convergence of such measures to the above stationary measures. We also employ hydrodynamics to provide further examples of convergence. http://front.math.ucdavis.edu/math.PR/0602098 --------------------------------------------------------------- 3994. BETA-COALESCENTS AND CONTINUOUS STABLE RANDOM TREES Julien Berestycki and Nathanael Berestycki and Jason Schweinsberg Coalescents with multiple collisions, also known as $\Lambda$- coalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case that the measure $\Lambda$ is the Beta$(2-\alpha,\alpha)$ distribution, they are also known to describe the genealogies of large populations where a single individual can produce a large number of offspring. Here we use a recent result of Birkner et al. to prove that Beta-coalescents can be embedded in continuous stable random trees, about which much is known due to recent progress of Duquesne and Le Gall. Our proof is based on a construction of the Donnelly-Kurtz lookdown process using continuous random trees which is of independent interest. This produces a number of results concerning the small-time behavior of Beta-coalescents. Most notably, we recover an almost sure limit theorem of the authors for the number of blocks at small times, and give the multifractal spectrum corresponding to the emergence of blocks with atypical size. Also, we are able to find exact asymptotics for sampling formulae corresponding to the site frequency spectrum and allele frequency spectrum associated with mutations in the context of population genetics. http://front.math.ucdavis.edu/math.PR/0602113 --------------------------------------------------------------- 3995. SAMPLE PATH LARGE DEVIATIONS FOR QUEUEING NETWORKS WITH BERNOULLI ROUTING Marc Lelarge This paper is devoted to the problem of sample path large deviations for multidimensional queueing models with feedback. We derive a new version of the contraction principle where the continuous map is not well-defined on the whole space: we give conditions under which it allows to identify the rate function. We illustrate our technique by deriving a large deviation principle for a class of networks that contains the classical Jackson networks. http://front.math.ucdavis.edu/math.PR/0602130 --------------------------------------------------------------- 3996. WASSERSTEIN DISTANCE ON CONFIGURATION SPACE L. Decreusefond We investigate here the optimal transportation problem on configuration space for the quadratic cost. It is shown that, as usual, provided that the corresponding Wasserstein is finite, there exists one unique optimal measure and that this measure is supported by the graph of the derivative (in the sense of the Malliavin calculus) of a ``concave'' (in a sense to be defined below) function. For finite point processes, we give a necessary and sufficient condition for the Wasserstein distance to be finite. http://front.math.ucdavis.edu/math.PR/0602134 --------------------------------------------------------------- 3997. SECOND ORDER ASYMPTOTICS FOR MATRIX MODELS Alice Guionnet (ENS Lyon - UMPA) and \'Edouard Maurel-Segala (ENS Lyon - UMPA) We study several-matrix models and show that when the potential is convex and a small perturbation of the Gaussian potential, the first order correction to the free energy can be expressed as a generating function for the enumeration of maps of genus one. In order to do that, we prove a central limit theorem for traces of words of the weakly interacting random matrices defined by these matrix models and show that the variance is a generating function for the number of planar maps with two vertices with prescribed colored edges. http://front.math.ucdavis.edu/math.PR/0601040 --------------------------------------------------------------- 3998. PERMUTATION TABLEAUX AND THE ASYMMETRIC EXCLUSION PROCESS Lauren K. Williams The partially asymmetric exclusion process (PASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites. It is partially asymmetric in the sense that the probability of hopping left is q times the probability of hopping right. In this paper we prove a close connection between the PASEP model and the combinatorics of permutation tableaux (certain 0-1 tableaux introduced in a previous paper with Steingrimsson). Namely, we prove that in the long time limit, the probability that the PASEP model is in a particular configuration tau is essentially the weight generating function for permutation tableaux of shape lambda(tau). The proof of this result uses a result of Derrida et al on the matrix ansatz for the PASEP. We derive a number of enumerative consequences of the connection between the PASEP model and permutation tableaux. One consequence is a generating function for the following (equidistributed) objects: the partition function for the PASEP model; permutation tableaux of length n+1, enumerated according to weight; permutations in S_{n+1}, enumerated according to crossings; permutations in S_{n+1}, enumerated according to occurrences of the generalized pattern 2-31. Another consequence is a generating function for the subset of the above objects which is specified by fixing (respectively) a configuration tau, a shape lambda(tau), a weak excedence set W(tau), or a descent set D(tau). Note that the equidistribution of permutation tableaux and permutations was proved in a previous paper of Steingrimsson and the author. http://front.math.ucdavis.edu/math.CO/0602109 --------------------------------------------------------------- 3999. NO-ARBITRAGE AND CLOSURE RESULTS FOR TRADING CONES WITH TRANSACTION COSTS Saul Jacka and Abdelkarem Berkaoui and Jon Warren The paper considers trading with proportional transaction costs. We give a necessary and sufficient condition for $A$, the cone of claims attainable from zero endowment, to be closed, and show, in general, how to represent its closure in such a way that it is the cone of claims attainable for zero endowment, for a different set of trading prices. The new representation obeys the Fundamental Theorem of Asset Pricing. We then show how to represent claims and in a final section show how any such setup corresponds to a coherent risk measure. http://front.math.ucdavis.edu/math.PR/0602178 --------------------------------------------------------------- 4000. ISOPERIMETRIC-TYPE INEQUALITIES FOR ITERATED BROWNIAN MOTION IN R^N Erkan Nane We extend generalized isoperimetric-type inequalities to iterated Brownian motion over several domains in $\RR{R}^{n}$. These kinds of inequalities imply in particular that for domains of finite volume, the exit distribution and moments of the first exit time for iterated Brownian motion are maximized with the ball $D^{*}$ centered at the origin, which has the same volume as $D$ http://front.math.ucdavis.edu/math.PR/0602188 --------------------------------------------------------------- 4001. CONDITIONING BY RARE SOURCES M. Grendar In this paper we study the exponential decay of posterior probability of a set of sources and conditioning by rare sources for both uniform and general prior distributions of sources. The decay rate is determined by L- divergence and rare sources from a convex, closed set asymptotically conditionally concentrate on an L-projection. L-projection on a linear family of sources belongs to Lambda-family of distributions. The results parallel those of Large Deviations for Empirical Measures (Sanov's Theorem and Conditional Limit Theorem). http://front.math.ucdavis.edu/math.ST/0601048 --------------------------------------------------------------- 4002. RANDOM SERIES OF FUNCTIONS AND APPLICATIONS Fr\'{e}d\'{e}ric Paccaut (LAMFA) and Dominique Schneider (LMPA) We study the continuity properties of trajectories for some random series of functions $\sum a\_kf(\alpha X\_k(\omega))$ where $a\_k$ is a complex sequence, $X\_k$ a sequence of real independent random variables, $f$ is a real valued function with period one and summable Fourier coefficients. We obtain almost sure continuity results for these periodic or almost periodic series for a large class of functions, where the "almost sure" does not depend on the function. http://front.math.ucdavis.edu/math.PR/0602207 --------------------------------------------------------------- 4003. QUANTUM STOCHATIC INTEGRALS AND DOOB-MEYER DECOMPOSITION Andrzej Luczak We show that for a quantum $L^p$-martingale $(X(t))$, $p>2$, there exists a Doob-Meyer decomposition of the submartingale $(|X(t)|^2)$. A noncommutative counterpart of a classical process continuous with probability one is introduced, and a quantum stochastic integral of such a process with respect to an $L^p$-martingale, $p>2$, is constructed. Using this construction, the uniqueness of the Doob-Meyer decomposition for a quantum martingale `continuous with probability one' is proved, and explicit forms of this decomposition and the quadratic variation process for such a martingale are obtained. http://front.math.ucdavis.edu/math.OA/0602216 --------------------------------------------------------------- 4004. LIMIT THEOREMS IN FREE PROBABILITY THEORY I G. P. Chistyakov and F. G\"otze Based on a new analytical approach to the definition of additive free convolution on probability measures on the real line we prove free analogs of limit theorems for sums for non-identically distributed random variables in classical Probability Theory. http://front.math.ucdavis.edu/math.OA/0602219 --------------------------------------------------------------- 4005. CRITICAL BRANCHING REGENERATIVE PROCESSES WITH MIGRATION George P. Yanev and Kosto V. Mitov and and Nickolay M. Yanev This paper demonstrates a new regeneration processes technology making use of positive stable distributions. We study the asymptotic behavior of branching processes with a randomly controlled migration component. Using the new method, we confirm some known results and establish new limit theorems that hold in a more general setting. http://front.math.ucdavis.edu/math.PR/0602261 --------------------------------------------------------------- 4006. A SUB-GAUSSIAN BERRY-ESSEEN THEOREM FOR THE HYPERGEOMETRIC DISTRIBUTION Soumendra N. Lahiri and A. Chatterjee and and T. Maiti In this paper, we derive a necessary and sufficient condition on the parameters of the Hypergeometric distribution for weak convergence to a Normal limit. We establish a Berry-Esseen theorem for the Hypergeometric distribution solely under this necessary and sufficient condition. We further derive a nonuniform Berry-Esseen bound where the tails of the difference between the Hypergeometric and the Normal distribution functions are shown to decay at a sub-Gaussian rate. http://front.math.ucdavis.edu/math.PR/0602276 --------------------------------------------------------------- 4007. RECOGNISING THE LAST RECORD OF A SEQUENCE Alexander Gnedin We study the best-choice problem for processes which generalise the process of records from Poisson-paced i.i.d. observations. Under the assumption that the observer knows distribution of the process and the horizon, we determine the optimal stopping policy and for a parametric family of problems also derive an explicit formula for the maximum probability of recognising the last record. http://front.math.ucdavis.edu/math.PR/0602278 --------------------------------------------------------------- 4008. BULK DIFFUSION OF 1D EXCLUSION PROCESS WITH BOND DISORDER A. Faggionato Given a doubly infinite sequence of positive numbers {c_k: k in Z} satisfying a LLN with limit A, we consider the nearest-neighbor simple exclusion process on Z where c_k is the probability rate of the jumps between k and k +1. If A is infinite we require an additional condition corresponding to macroscopic homogeneity of the medium. By extending a method developed by K. Nagy we show that the diffusively rescaled process has hydrodynamic behavior described by the heat equation with diffusion constant 1/A. In particular, the process has diffusive behavior for finite A and subdiffusive behavior for infinite A. http://front.math.ucdavis.edu/math.PR/0601076 --------------------------------------------------------------- 4009. AR(1) SCHEMES WITH SEMI-STABLE MARGINALS S Satheesh and E Sandhya The family of semi-stable laws is shown to be infinitely divisible and semi-selfdecomposable. Thus they qualify to model AR(1) schemes. The structure of AR(1) schemes with semi-stable marginals are explored. http://front.math.ucdavis.edu/math.PR/0602286 --------------------------------------------------------------- 4010. INVARIANCE PRINCIPLES FOR RANDOM WALKS CONDITIONED TO STAY POSITIVE Francesco Caravenna and Lo\"ic Chaumont Let {S_n} be a random walk in the domain of attraction of a stable law Y, i.e. there exists a sequence of positive real numbers (a_n) such that S_n/a_n converges in law to Y. Our main result is that the rescaled process (S_[nt]/a_n, t \ge 0), when conditioned to stay positive for all the time, converges in law (in the functional sense) towards the corresponding stable Levy process conditioned to stay positive in the same sense. Under some additional assumptions, we also prove a related invariance principle for the random walk killed at its first entrance in the negative half-line and conditioned to die at zero. http://front.math.ucdavis.edu/math.PR/0602306 --------------------------------------------------------------- 4011. A PARRONDO'S PARADOX IN RELIABILITY THEORY Antonio Di Crescenzo Parrondo's paradox arises in sequences of games in which a winning expectation may be obtained by playing the games in a random order, even though each game in the sequence may be lost when played individually. We present a suitable version of Parrondo's paradox in reliability theory involving two systems in series, the units of the first system being less reliable than those of the second. If the first system is modified so that the distributions of its new units are mixtures of the previous distributions with equal probabilities, then under suitable conditions the new system is shown to be more reliable than the second in the "usual stochastic order" sense. http://front.math.ucdavis.edu/math.PR/0602308 --------------------------------------------------------------- 4012. ON PERMANENTAL POLYNOMIALS OF CERTAIN RANDOM MATRICES Yan V Fyodorov The paper addresses the calculation of correlation functions of permanental polynomials of matrices with random entries. By exploiting a convenient contour integral representation of the matrix permanent some explicit results are provided for several random matrix ensembles. When compared with the corresponding formulae for characteristic polynomials, our results show both striking similarities and interesting differences. Based on these findings, we conjecture the asymptotic forms of the density of permanental roots in the complex plane for Gaussian ensembles as well as for the Circular Unitary Ensemble of large matrix dimension. http://front.math.ucdavis.edu/math-ph/0602039 --------------------------------------------------------------- 4013. DETERMINISTIC RANDOM WALKS ON THE INTEGERS Joshua Cooper and Benjamin Doerr and Joel Spencer and and Gabor Tardos Jim Propp's P-machine, also known as the "rotor router model" is a simple deterministic process that simulates a random walk on a graph. Instead of distributing chips to randomly chosen neighbors, it serves the neighbors in a fixed order. We investigate how well this process simulates a random walk. For the graph being the infinite path, we show that, independent of the starting configuration, at each time and on each vertex, the number of chips on this vertex deviates from the expected number of chips in the random walk model by at most a constant c_1, which is approximately 2.29. For intervals of length L, this improves to a difference of O(log L), for the L_2 average of a contiguous set of intervals even to O(sqrt{log L}). All these bounds are tight. http://front.math.ucdavis.edu/math.CO/0602300 --------------------------------------------------------------- 4014. FILTRATION-CONSISTENT DYNAMIC OPERATOR WITH A FLOOR AND ASSOCIATED REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS Xiaobo Bao and Shanjian Tang This paper introduces the notion of a filtration-consistent dynamic operator with a floor, by suitably formulating four axioms. It is shown that under some suitable conditions, a filtration-consistent dynamic operator with a continuous upper-bounded floor is necessarily represented by the solution of a backward stochastic differential equation reflected upwards on the floor. http://front.math.ucdavis.edu/math.PR/0602322 --------------------------------------------------------------- 4015. DUAL REPRESENTATION AS STOCHASTIC DIFFERENTIAL GAMES OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND DYNAMIC EVALUATIONS Shanjian Tang In this Note, assuming that the generator is uniform Lipschitz in the unknown variables, we relate the solution of a one dimensional backward stochastic differential equation with the value process of a stochastic differential game. Under a domination condition, a filtration-consistent evaluations is also related to a stochastic differential game. This relation comes out of a min-max representation for uniform Lipschitz functions as affine functions. The extension to reflected backward stochastic differential equations is also included. http://front.math.ucdavis.edu/math.PR/0602323 --------------------------------------------------------------- 4016. A NOTE ON THE CONNECTION BETWEEN MOLCHAN-GOLOSOV- AND MANDELBROT-VAN NESS REPRESENTATION OF FRACTIONAL BROWNIAN MOTION Celine Jost (University of Helsinki) We proof a connection between the generalized Molchan-Golosov integral transform and the generalized Mandelbrot-Van Ness integral transform of fractional Brownian motion (fBm). The former changes fBm of arbitrary Hurst index K into fBm of index H by integrating over [0,t], whereas the latter requires integration over (-infty,t]. http://front.math.ucdavis.edu/math.PR/0602356 --------------------------------------------------------------- 4017. EULER ESTIMATES OF ROUGH DIFFERENTIAL EQUATIONS Peter Friz and Nicolas Victoir We consider controlled differential equations and give new estimates for higher order Euler schemes. Our proofs are inspired by recent work of A. M. Davie who considers first and second order schemes. In order to implement the general case we make systematic use of geodesic approximations in the free nilpotent group. As application, we can control moments of solutions to rough path differential equations (RDEs) driven by random rough paths with sufficient integrability and have a criteria for L^q - convergence in the Universal Limit Theorem. We also obtain Azencott type estimates and asymptotic expansions for random RDE solution. When specialized to RDEs driven by Enhanced Brownian motion, we (mildly) improve classic estimates for diffusions in the small time limit. http://front.math.ucdavis.edu/math.CA/0602345 --------------------------------------------------------------- 4018. COMPLEX ANALYSIS METHODS IN NONCOMMUTATIVE PROBABILITY Serban Teodor Belinschi In this thesis we study convolutions that arise from noncommutative probability theory. We prove several regularity results for free convolutions, and for measures in partially defined one-parameter free convolution semigroups. We discuss connections between Boolean and free convolutions and, in the last chapter, we prove that any infinitely divisible probability measure with respect to monotonic additive or multiplicative convolution belongs to a one-parameter semigroup with respect to the corresponding convolution. Earlier versions of some of the results in this thesis have already been published, while some others have been submitted for publication. We have preserved almost entirely the specific format for PhD theses required by Indiana University. This adds several unnecessary pages to the document, but we wanted to preserve the specificity of the document as a PhD thesis at Indiana University. http://front.math.ucdavis.edu/math.OA/0602343 --------------------------------------------------------------- 4019. OPTIMALLY COUPLING THE KOLMOGOROV DIFFUSION, AND RELATED OPTIMAL CONTROL PROBLEMS Kalvis M. Jansons and Paul D. Metcalfe We discuss the optimal Markovian coupling before an exponential time of the Kolmogorov diffusion, and a class of related stochastic control problems in which the aim is to hit the origin before an exponential time. We provide a scaling argument for the optimal control in the near field and use rational WKB approximation to obtain the optimal control in the far field, and compare these analytical results with numerical experiments. In some of these optimal control problems, in which the advection velocity field is bounded, we show that the probability of success field agrees exactly with its leading-order asymptotic approximation in some areas of the plane, up to an undetermined multiplicative constant. We conjecture a necessary and sufficient condition for this behaviour, which is strongly supported by numerical experiments. http://front.math.ucdavis.edu/math.PR/0602365 --------------------------------------------------------------- 4020. STOCHASTIC GENERALIZED POROUS MEDIA AND FAST DIFFUSION EQUATIONS Jiagang Ren and Michael R\"ockner and Feng-Yu Wang We present a generalization of Krylov-Rozovskii's result on the existence and uniqueness of solutions to monotone stochastic differential equations. As an application, the stochastic generalized porous media and fast diffusion equations are studied for $\sigma$-finite reference measures, where the drift term is given by a negative definite operator acting on a time-dependent function, which belongs to a large class of functions comparable with the so-called $N$-functions in the theory of Orlicz spaces. http://front.math.ucdavis.edu/math.PR/0602369 --------------------------------------------------------------- 4021. DISTORTION MISMATCH IN THE QUANTIZATION OF PROBABILITY MEASURES Siegried Graf (Universit\"{a}t Passau) and Harald Luschgy and Gilles Pag\`es (PMA) We elucidate the asymptotics of the L^s-quantization error induced by a sequence of L^r-optimal n-quantizers of a probability distribution P on R^d when s>r. In particular we show that under natural assumptions, the optimal rate is preserved as long as s{1/2}$. In particular, we show, using stochastic calculus, that this process admits a skew-product decomposition and deduce from this representation some asymptotic properties of the motion. http://front.math.ucdavis.edu/math.PR/0602547 --------------------------------------------------------------- 4055. LOGARITHMIC SOBOLEV INEQUALITIES FOR INHOMOGENEOUS MARKOV SEMIGROUPS Jean-Fran\c{c}ois Collet (JAD) and Florent Malrieu (IRMAR) We investigate the dissipativity properties of a class of scalar second order parabolic partial differential equations with time-dependent coefficients. We provide explicit condition on the drift term which ensure that the relative entropy of one particular orbit with respect to some other one decreases to zero. The decay rate is obtained explicitly by the use of a Sobolev logarithmic inequality for the associated semigroup, which is derived by an adaptation of Bakry's $\Gamma-$ calculus. As a byproduct, the systematic method for constructing entropies which we propose here also yields the well-known intermediate asymptotics for the heat equation in a very quick way, and without having to rescale the original equation. http://front.math.ucdavis.edu/math.PR/0602548 --------------------------------------------------------------- 4056. EXACT INEQUALITIES FOR SUMS OF ASYMMETRIC RANDOM VARIABLES, WITH APPLICATIONS Iosif Pinelis Let $\BS_1,...,\BS_n$ be independent identically distributed random variables each having the standardized Bernoulli distribution with parameter $p \in(0,1)$. Let $m_*(p):=(1+p+2p^2)/(2\sqrt{p-p^2}+4p^2)$ if $02/3$ it is shown that there is exponential decay of correlations. http://front.math.ucdavis.edu/math.PR/0601157 --------------------------------------------------------------- 4068. CGMY AND MEIXNER SUBORDINATORS ARE ABSOLUTELY CONTINUOUS WITH RESPECT TO ONE SIDED STABLE SUBORDINATORS Dilip Madan and Marc Yor (PMA) We describe the CGMY and Meixner processes as time changed Brownian motions. The CGMY uses a time change absolutely continuous with respect to the one-sided stable $(Y/2)$ subordinator while the Meixner time change is absolutely continuous with respect to the one sided stable $(1/2)$ subordinator$. $ The required time changes may be generated by simulating the requisite one-sided stable subordinator and throwing away some of the jumps as described in Rosinski (2001). http://front.math.ucdavis.edu/math.PR/0601173 --------------------------------------------------------------- 4069. PERFECT SIMULATION FOR A CLASS OF POSITIVE RECURRENT MARKOV CHAINS Stephen Connor and Wilfrid Kendall This paper generalises the work of Kendall (Electronic Communications in Probability 2004, vol 9, 140-151), which showed that perfect simulation, in the form of dominated coupling from the past, is always possible (though not necessarily practical) for geometrically ergodic Markov chains. Here we consider the more general situation of positive recurrent chains, and explore when it is possible to produce such a simulation algorithm for these chains. We introduce a class of chains which we name "tame", for which we show that perfect simulation is possible. http://front.math.ucdavis.edu/math.PR/0601174 --------------------------------------------------------------- 4070. NUMBER VARIANCE OF RANDOM ZEROS Bernard Shiffman and Steve Zelditch The main results of this article are asymptotic formulas for the variance of the number of zeros of a Gaussian random polynomial of degree $N$ in an open set $U \subset \C$ as the degree $N \to \infty$, and more generally for the zeros of random holomorphic sections of high powers of any positive line bundle over any Riemann surface. The formulas were conjectured in special cases by Forrester and Honner. In higher dimensions, we give similar formulas for the variance of the volume inside a domain $U$ of the zero hypersurface of a random holomorphic section of a high power of a positive line bundle over any compact K\"ahler manifold. These results generalize the variance asymptotics of Sodin and Tsirelson for special model ensembles of chaotic analytic functions in one variable to any ample line bundle and Riemann surface. We also combine our methods with those of Sodin-Tsirelson to generalize their asymptotic normality results for smoothed number statistics. http://front.math.ucdavis.edu/math.CV/0512652 --------------------------------------------------------------- 4071. THE LOWER ENVELOPE OF POSITIVE SELF-SIMILAR MARKOV PROCESSES Lo\"{i}c Chaumont (PMA) and Juan-Carlos Pardo (PMA) We establish integral tests and laws of the iterated logarithm for the lower envelope of positive self-similar Markov processes at 0 and $+\infty $. Our proofs are based on the Lamperti representation and time reversal arguments. These results extend laws of the iterated logarithm for Bessel processes due to Dvoretsky and Erd\"{o}s, Motoo and Rivero. http://front.math.ucdavis.edu/math.PR/0601177 --------------------------------------------------------------- 4072. A LOG-SOBOLEV TYPE INEQUALITY FOR FREE ENTROPY OF TWO PROJECTIONS Fumio Hiai and Yoshimichi Ueda We prove an inequality between the free entropy and the mutual free Fisher information for two projections, regarded as a free analog of the logarithmic Sobolev inequality. The proof is based on the random matrix approximation procedure via the Grassmannian random matrix model of two projections. http://front.math.ucdavis.edu/math.OA/0601171 --------------------------------------------------------------- 4073. RECURRENCE AND TRANSIENCE OF EXCITED RANDOM WALKS ON $\Z^D$ AND STRIPS Martin P.W. Zerner We investigate excited random walks on $\Z^d, d\ge 1,$ and on planar strips $\Z\times\{0,1,...,L-1\}$ which have a drift in a given direction. The strength of the drift may depend on a random i.i.d. environment and on the local time of the walk. We give exact criteria for recurrence and transience, thus generalizing results by Benjamini and Wilson for once-excited random walk on $\Z^d$ and by the author for multi-excited random walk on $\Z$. http://front.math.ucdavis.edu/math.PR/0601233 --------------------------------------------------------------- 4074. INFINITE-DIMENSIONAL QUADRATURE AND QUANTIZATION Steffen Dereich and Thomas Mueller-Gronbach and Klaus Ritter We study numerical integration of Lipschitz functionals on a Banach space by means of deterministic and randomized (Monte Carlo) algorithms. This quadrature problem is shown to be closely related to the problem of quantization of the underlying probability measure. In addition to the general setting we analyze in particular integration w.r.t. Gaussian measures and distributions of diffusion processes. We derive lower bounds for the worst case error of every algorithm in terms of its computational cost, and we present matching upper bounds, up to logarithms, and corresponding almost optimal algorithms. As auxiliary results we determine the asymptotic behaviour of quantization numbers and Kolmogorov widths for diffusion processes. http://front.math.ucdavis.edu/math.PR/0601240 --------------------------------------------------------------- 4075. ON SOME TRANSFORMATIONS BETWEEN POSITIVE SELF--SIMILAR MARKOV PROCESSES Lo\"{i}c Chaumont (PMA) and V\'{i}ctor Manuel Rivero (CIMAT) A path decomposition at the infimum for positive self-similar Markov processes (pssMp) is obtained. Next, several aspects of the conditioning to hit 0 of a pssMp are studied. Associated to a given a pssMp $X,$ that never hits 0, we construct a pssMp $X^{\downarrow}$ that hits 0 in a finite time. The latter can be viewed as $X$ conditioned to hit 0 in a finite time and we prove that this conditioning is determined by the pre-minimum part of $X.$ Finally, we provide a method for conditioning a pssMp that hits 0 by a jump to do it continuously. http://front.math.ucdavis.edu/math.PR/0601243 --------------------------------------------------------------- 4076. FREE REAL EXPONENTIAL FAMILIES Wlodzimierz Bryc Following the analogy with classical reproductive exponential models, we study the properties of free exponential families. http://front.math.ucdavis.edu/math.PR/0601273 --------------------------------------------------------------- 4077. THE FEYNMAN GRAPH REPRESENTATION OF CONVOLUTION SEMIGROUPS AND ITS APPLICATIONS TO LEVY STATISTICS H. Gottschalk and B. Smii and H. Thaler We consider the Cauchy problem for a pseudo differential operator which has a translation invariant and analytic symbol. For a certain set of initial conditions, a formal solution is obtained by a perturbative expansion. The so-obtained series can be re-expressed in terms of generalized Feynman graphs and Feynman rules. The logarithm of the solution then can be represented by a series containing the connected Feynman graphs, only. Under some conditions, it is shown that the formal solution uniquely determines the real solution by the means of Borel transforms. The formalism is then applied to probabilistic Levy distributions. Here, the Gaussian part of such a distribution is re-interpreted as a initial condition, and a large diffusion expansion for L\'evy densities is obtained. It is outlined, how this expansion can be used in statistical problems that involve Levy distributions. http://front.math.ucdavis.edu/math.PR/0601278 --------------------------------------------------------------- 4078. STOCHASTIC NETWORKS WITH MULTIPLE STABLE POINTS Nelson Antunes (UAL) and Christine Fricker (INRIA Rocquencourt) and Philippe Robert (INRIA Rocquencourt), Danielle Tibi (PMA) This paper analyzes stochastic networks consisting of a set of finite capacity sites where different classes of individuals move according to some routing policy. The associated (non-reversible) Markov jump processes are analyzed under a thermodynamic limit regime, i.e. when the networks have some symmetry properties and when the number of nodes goes to infinity. A metastability property is proved: under some conditions on the parameters, it is shown that, in the limit, several equilibrium points coexist for the empirical distribution. The key ingredient of the proof of this property is a dimension reduction achieved by the introduction of two energy functions and a convenient mapping of their local minima and saddle points. Cases with a unique equilibrium point are also presented. http://front.math.ucdavis.edu/math.PR/0601296 --------------------------------------------------------------- 4079. CORRELATED EQUILIBRIA IN COMPETITIVE STAFF SELECTION PROBLEM David M. Ramsey and Krzysztof Szajowski This paper deals with an extension of the concept of correlated strategies to Markov stopping games. The Nash equilibrium approach to solving nonzero-sum stopping games may give multiple solutions. An arbitrator can suggest to each player the decision to be applied at each stage based on a joint distribution over the players' decisions. This is a form of equilibrium selection. Examples of correlated equilibria in nonzero-sum games related to the staff selection competition in the case of two departments are given. Utilitarian, egalitarian, republican and libertarian concepts of correlated equilibria selection are used. http://front.math.ucdavis.edu/math.OC/0601289 --------------------------------------------------------------- 4080. VARIATIONS OF THE SOLUTION TO A STOCHASTIC HEAT EQUATION Jason Swanson We consider the solution to a stochastic heat equation. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time, F(t), has a nontrivial quartic variation. This process, therefore, has infinite quadratic variation and is not a semimartingale. It follows that the classical Ito calculus does not apply. Motivated by heuristic ideas about a possible new calculus for this process, we are led to study modifications of the quadratic variation. Namely, we modify each term in the sum of the squares of the increments so that it has mean zero. We then show that these sums, as functions of t, converge weakly to Brownian motion. http://front.math.ucdavis.edu/math.PR/0601007 --------------------------------------------------------------- 4081. CONSERVATIVE STOCHASTIC CAHN-HILLIARD EQUATION WITH REFLECTION Arnaud Debussche and Lorenzo Zambotti We consider a stochastic partial differential equation with reflection at 0 and with the constraint of conservation of the space average. The equation is driven by the derivative in space of a space-time white noise and contains a double Laplacian in the drift. Due to the lack of the maximum principle for the double Laplacian, the standard techniques based on the penalization method do not yield existence of a solution. We propose a method based on infinite dimensional integration by parts formulae, obtaining existence and uniqueness of a strong solution for all continuous non-negative initial conditions and detailed information on the associated invariant measure and Dirichlet Form. http://front.math.ucdavis.edu/math.PR/0601313 --------------------------------------------------------------- 4082. RECENT ADVANCES IN INVARIANCE PRINCIPLES FOR STATIONARY SEQUENCES Florence Merlevede and Magda Peligrad and Sergey Utev In this paper we survey some recent results on the central limit theorem and its weak invariance principle for stationary sequences. We also describe several maximal inequalities that are the main tool for obtaining the invariance principles, and also they have interest in themselves. The classes of dependent random variables considered will be martingale-like sequences, mixing sequences, linear processes, additive functionals of ergodic Markov chains. http://front.math.ucdavis.edu/math.PR/0601315 --------------------------------------------------------------- 4083. HEAT KERNEL MEASURE ON CENTRAL EXTENSION OF CURRENT GROUPS IN ANY DIMENSION Remi Leandre We define measures on central extension of current groups in any dimension by using infinite dimensional Brownian motion. http://front.math.ucdavis.edu/math.PR/0601330 --------------------------------------------------------------- 4084. CRITICAL GALTON-WATSON PROCESSES: THE MAXIMUM OF TOTAL PROGENIES WITHIN A LARGE WINDOW Klaus Fleischmann and Vladimir A. Vatutin and Vitali Wachtel Consider a critical Galton-Watson process Z={Z_n: n=0,1,...} of index 1+alpha, alpha in (0,1]. Let S_k(j) denote the sum of the Z_n with n in the window [k,...,k+j), and M_m(j) the maximum of the S_k with k moving in [0,m-j]. We describe the asymptotic behavior of the expectation EM_m(j) if the window width j=j_m is such that j/m converges in [0,1] as m tends to infinity. This will be achieved via establishing the asymptotic behavior of the tail probabilities of M_{infinity}(j). http://front.math.ucdavis.edu/math.PR/0601333 --------------------------------------------------------------- 4085. DIFFERENCES BETWEEN INDEPENDENT VARIABLES AND ALMOST BENFORD BEHAVIOR Steven J. Miller and Mark. J. Nigrini Fix a base B and let X_1, ..., X_N be independent identically distributed random variables. If the X_i's are drawn from a uniform distribution, then as N tends to infinity the distribution of the digits of the differences between adjacent X_i's tends to a universal distribution which is almost Benford's Law; we call this Almost Benford behavior. For each base we develop a rapidly convergent Fourier series expansion. In base e one term yields five digits of accuracy; in base 10 two terms yield three digits. Fix a \delta in (0,1) and choose N independent random variables from a nice probability density. The distribution of digits of any N^\delta consecutive differences and all N-1 normalized differences of the X_i's exhibit Almost Benford behavior. We derive conditions on the probability density which determine whether or not the distribution of the digits of all the un-normalized differences converges to Benford's Law, Almost Benford behavior, or oscillates between the two. As an example the Pareto distribution leads to oscillating behavior. We introduce a new technique to study equidistribution questions modulo 1; such questions have long been known to be related to Benford's Law. By differentiating the cumulative distribution function of the logarithms modulo 1, applying Poisson Summation and then integrating the resulting expression, we derive rapidly converging explicit formulas measuring the deviations from Benford's Law. http://front.math.ucdavis.edu/math.PR/0601344 --------------------------------------------------------------- 4086. IFSM REPRESENTATION OF BROWNIAN MOTION WITH APPLICATIONS TO SIMULATION S. M. Iacus and D. La Torre Several methods are currently available to simulate paths of the Brownian motion. In particular, paths of the BM can be simulated using the properties of the increments of the process like in the Euler scheme, or as the limit of a random walk or via L2 decomposition like the Kac-Siegert/Karnounen-Loeve series. In this paper we first propose a IFSM (Iterated Function Systems with Maps) operator whose fixed point is the trajectory of the BM. We then use this representation of the process to simulate its trajectories. The resulting simulated trajectories are self-affine, continuous and fractal by construction. This fact produces more realistic trajectories than other schemes in the sense that their geometry is closer to the one of the true BM's trajectories. The IFSM trajectory of the BM can then be used to generate more realistic solutions of stochastic differential equations. http://front.math.ucdavis.edu/math.PR/0601379 --------------------------------------------------------------- 4087. INTEGRAL CRITERIA FOR TRANSPORTATION-COST INEQUALITIES Nathael Gozlan (MODAL'X) In this paper, we provide a characterization of a large class of transportation-cost inequalities in terms of exponential integrability of the cost function under the reference probability measure. Our results completely extend the previous works by Djellout, Guilin and Wu and Bolley and Villani. http://front.math.ucdavis.edu/math.PR/0601384 --------------------------------------------------------------- 4088. THE POLYNOMIAL METHOD FOR RANDOM MATRICES N. Raj Rao and Alan Edelman We define a class of "algebraically characterizable" random matrices. These are random matrices for which the Stieltjes transform of the limiting spectral measure is an algebraic function. The famous semi-circle law for Wigner matrices and the Marcenko-Pastur law for Wishart matrices are special cases. The practical utility of this definition can be succinctly summarized: if a random matrix is shown to be algebraic then its limiting spectral measure can be computed using a simple root-finding algorithm. Furthermore, if the moments exist, then the corresponding moment generating function will be differentiably finite so that we will often be able to enumerate them efficiently in closed form. Algebraicity of a random matrix acts as a certificate of the computability of its limiting spectral measure and moments. We specify the class of such random matrices by its generators and demonstrate that the transforms of "free probability" that encode free additive and multiplicative convolution can be expressed as bivariate resultants. We present a simple computational realization, a random matrix "calculator" as it were, based on the "polynomial method" that finally allows researchers to harness the power of free probability and infinite random matrix theory. http://front.math.ucdavis.edu/math.PR/0601389 --------------------------------------------------------------- 4089. A LARGE DEVIATION PRINCIPLE FOR JOIN THE SHORTEST QUEUE Anatolii A. Puhalskii and Alexander A. Vladimirov We consider a join-the-shortest-queue model which is as follows. There are $K$ single FIFO servers and $M$ arrival processes. The customers from a given arrival process can be served only by servers from a certain subset of all servers. The actual destination is the server with the smallest weighted queue length. The arrival processes are assumed to obey a large deviation principle while the service is exponential. A large deviation principle is established for the queue-length process. The action functional is expressed in terms of solutions to mathematical programming problems. The large deviation limit point is identified as a weak solution to a system of idempotent equations. Uniqueness of the weak solution is proved by establishing trajectorial uniqueness. http://front.math.ucdavis.edu/math.PR/0601010 --------------------------------------------------------------- 4090. ON ALMOST-SURE VERSIONS OF CLASSICAL LIMIT THEOREMS FOR DYNAMICAL SYSTEMS J-R Chazottes and S Gouezel The purpose of this article is to construct a toolbox, in Dynamical Systems, to support the idea that ``whenever we can prove a limit theorem in the classical sense for a dynamical system, we can prove a suitable almost-sure version based on an empirical measure with log-average''. We follow three different approaches: martingale methods, spectral methods and induction arguments. Our results apply among others to Axiom A maps or flows, and to systems inducing a Gibbs-Markov map. http://front.math.ucdavis.edu/math.DS/0601388 --------------------------------------------------------------- 4091. REVERSALS OF CHANCE IN PARADOXICAL GAMES P. Amengual and P. Meurs and B. Cleuren and R. Toral We present two collective games with new paradoxical features when they are combined. Besides reproducing the so--called Parrondo effect, where a winning game is obtained from the alternation of two fair games, a new effect appears, i.e., there exists a current inversion when varying the mixing probability between the games. We present a detailed study by means of a discrete--time Markov chain analysis, obtaining analytical expressions for the stationary probabilities for a finite number of players. We also provide some qualitatively insight into this new current inversion effect. http://front.math.ucdavis.edu/math.PR/0601404 --------------------------------------------------------------- 4092. RANDOM WALKS AND POLYMERS IN THE PRESENCE OF QUENCHED DISORDER Cecile Monthus After a general introduction to the field, we describe some recent results concerning disorder effects on both `random walk models', where the random walk is a dynamical process generated by local transition rules, and on `polymer models', where each random walk trajectory representing the configuration of a polymer chain is associated to a global Boltzmann weight. For random walk models, we explain, on the specific examples of the Sinai model and of the trap model, how disorder induces anomalous diffusion, aging behaviours and Golosov localization, and how these properties can be understood via a strong disorder renormalization approach. For polymer models, we discuss the critical properties of various delocalization transitions involving random polymers. We first summarize some recent progresses in the general theory of random critical points : thermodynamic observables are not self-averaging at criticality whenever disorder is relevant, and this lack of self-averaging is directly related to the probability distribution of pseudo-critical temperatures $T_c(i,L)$ over the ensemble of samples $(i)$ of size $L$. We describe the results of this analysis for the bidimensional wetting and for the Poland-Scheraga model of DNA denaturation. http://front.math.ucdavis.edu/cond-mat/0601332 --------------------------------------------------------------- 4093. HOROCYCLIC PRODUCTS OF TREES Laurent Bartholdi and Markus Neuhauser and Wolfgang Woess Let T_1,..., T_d be homogeneous trees with degrees q_1+1,..., q_d+1>=3, respectively. For each tree, let h:T_j->Z be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of T_1,...,T_d is the graph DL(q_1,...,q_d) consisting of all d-tuples x_1...x_d in T_1x...xT_d with h(x_1)+...+h(x_d)=0, equipped with a natural neighbourhood relation. In the present paper, we explore the geometric, algebraic, analytic and probabilistic properties of these graphs and their isometry groups. If d=2 and q_1=q_2=q then we obtain a Cayley graph of the lamplighter group (wreath product) (Z/qZ) wr Z. If d=3 and q_1=q_2=q_3=q then DL is the Cayley graph of a finitely presented group into which the lamplighter group embeds naturally. Also when d>=4 and q_1=...=q_d=q is such that each prime power in the decomposition of q is larger than d-1, we show that DL is a Cayley graph of a finitely presented group. This group is of type F_{d-1}, but not F_d. It is not automatic, but it is an automata group in most cases. On the other hand, when the q_j do not all coincide, DL (q_1,...,q_d) is a vertex-transitive graph, but is not the Cayley graph of a finitely generated group. Indeed, it does not even admit a group action with finitely many orbits and finite point stabilizers. The l^2-spectrum of the ``simple random walk'' operator on DL is always pure point. When d=2, it is known explicitly from previous work, while for d=3 we compute it explicitly. Finally, we determine the Poisson boundary of a large class of group-invariant random walks on DL. It coincides with a part of the geometric boundary of DL. http://front.math.ucdavis.edu/math.CO/0601417 --------------------------------------------------------------- 4094. LARGE DEVIATIONS FOR NON-UNIFORMLY EXPANDING MAPS V Araujo M J Pacifico We obtain large deviation results for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average decays to zero exponentially fast with the number of iterates involved. As easy by-products we deduce escape rates from subsets of the basins of physical measures for these types of maps. http://front.math.ucdavis.edu/math.DS/0601449 --------------------------------------------------------------- 4095. THE SUBMARTINGALE PROBLEM FOR A CLASS OF DEGENERATE ELLIPTIC OPERATORS Richard F. Bass and Alexander Lavrentiev We consider the degenerate elliptic operator acting on $C^2$ functions on $[0,\infty)^d$: \[ L f(x)=\sum_{i=1}^d a_i(x) x_i^{\alpha_i} \frac {\partial^2 f}{\partial x_i^2} (x) +\sum_{i=1}^d b_i(x) \frac{\partial f} {\partial x_i}(x), \] where the $a_i$ are continuous functions that are bounded above and below by positive constants, the $b_i$ are bounded and measurable, and the $ \alpha_i\in (0,1)$. We impose Neumann boundary conditions on the boundary of $[0,\infty)^d$. There will not be uniqueness for the submartingale problem corresponding to $L$. If we consider, however, only those solutions to the submartingale problem for which the process spends 0 time on the boundary, then existence and uniqueness for the submartingale problem for $L$ holds within this class. Our result is equivalent to establishing weak uniqueness for the system of stochastic differential equations \[ dX_t^i=\sqrt{2a_i(X_t)} (X_t^i)^{\alpha_i/2} dW^i_t+b_i(X_t) dt +dL_t^{X^i}, where X^i_t\geq 0, \] where $W_t^i$ are independent Brownian motions and $L^{X_i}_t$ is a local time at 0 for $X^i$. http://front.math.ucdavis.edu/math.PR/0601027 --------------------------------------------------------------- 4096. HOW MANY ENTRIES OF A TYPICAL ORTHOGONAL MATRIX CAN BE APPROXIMATED BY INDEPENDENT NORMALS? Tiefeng Jiang We solve an open problem of Diaconis that asks what are the largest orders of p_n and q_n such that Z_n, the p_n\times q_n upper left block of a random matrix \bold{\Gamma}_n which is uniformly distributed on the orthogonal group O(n), can be approximated by independent standard normals? This problem is solved by two different approximation methods. First, we show that the {\it variation distance} between the joint distribution of entries of Z_n and that of p_nq_n independent standard normals goes to zero provided p_n=o(\sqrt{n}) and q_n=o(\sqrt{n}). We also show that the above variation distance does not go to zero if p_n=[x\sqrt{n}] and q_n=[y\sqrt{n}] for any positive numbers x and y. This says that the largest orders of p_n and q_n are o(n^{1/2}) in the sense of the above approximation. Second, suppose \bold{\Gamma}_n=(\gamma_{ij})_{n\times n} is generated by performing the Gram-Schmidt algorithm on the columns of \bold{Y}_n=(y_{ij})_{n\times n} where \{y_{ij}; 1\leq i, j \leq n\} are i.i.d. standard normals. We show that \epsilon_n(m):=\max_{1\leq i \leq n, 1 \leq j \leq m}|\sqrt{n}\gamma_{ij}-y_{ij}| goes to zero in probability as long as m=m_n=o(n/\log n). We also prove that \epsilon_n(m_n)\to 2\sqrt {\alpha} in probability when m_n=[n\alpha/\log n] for any \alpha>0. This says that m_n=o(n/\log n) is the largest order such that the entries of the first m_n columns of \bold{\Gamma}_n can be approximated simultaneously by independent standard normals. http://front.math.ucdavis.edu/math.PR/0601457 --------------------------------------------------------------- 4097. ERROR BOUNDS FOR AMERICAN PUT OPTION PRICING BASED ON "NON- RECOMBINING" TREES Frederik S Herzberg Consider a discrete finite-dimensional, Markovian market model. In this setting, discretely sampled American options can be priced using the so-called ``non-recombining'' tree algorithm. By successively increasing the number of exercise times, one gets more ``realistic'' approximations to the American option price. For combinatorial reasons, we shall consider a recursive algorithm that doubles the number of exercise times at each recursion step. First we prove, by elementary arguments, error bounds for the first order differences in this recursive algorithm. From this, bounds on the higher order differences can be obtained using combinatorial arguments that are motivated by the theory of rough paths. We shall obtain an explicit $L^1(C)$ convergence estimate for the recursive algorithm that prices a discretely sampled American $max$-put option (on a basket of size $d$) at each recursion step, $C $ being a compact subset of $\RR^d$, under the assumption of sufficiently small volatilities. http://front.math.ucdavis.edu/math.PR/0601468 --------------------------------------------------------------- 4098. ISOPERIMETRY BETWEEN EXPONENTIAL AND GAUSSIAN Franck Barthe (LSProba) and Patrick Cattiaux (CMAP and MODAL'X) and Cyril Roberto (LAMA) We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate half-spaces are approximate solutions of the isoperimetric problem. http://front.math.ucdavis.edu/math.PR/0601475 --------------------------------------------------------------- 4099. ANNEALED TAIL ESTIMATES FOR A BROWNIAN MOTION IN A DRIFTED BROWNIAN POTENTIAL Marina Talet We study Brownian motion in a drifted Brownian potential in the subexponential regime. We prove that the annealed probability of deviating below the almost sure speed has a polynomial rate of decay and compute the exponent in this power law. This provides a continuous-time analogue of what Dembo, Peres and Zeitouni proved for the transient random walk in random environment. Our method takes a completely different route, making use of Lamperti's representation together with an iteration scheme. http://front.math.ucdavis.edu/math.PR/0601500 --------------------------------------------------------------- 4100. PATHWISE UNIQUENESS FOR A DEGENERATE STOCHASTIC DIFFERENTIAL EQUATION Richard F. Bass and Krzysztof Burdzy and and Zhen-Qing Chen We introduce a new method of proving pathwise uniqueness, and we apply it to the degenerate stochastic differential equation $$dX_t=|X_t|^\alpha dW_t,$$ where $W_t$ is a one-dimensional Brownian motion and $\alpha\in(0,1/2) $. Weak uniqueness does not hold for the solution to this equation. If one restricts attention, however, to those solutions that spend zero time at 0, then pathwise uniqueness does hold and a strong solution exists. We also consider a class of stochastic differential equations with reflection. http://front.math.ucdavis.edu/math.PR/0601505 --------------------------------------------------------------- 4101. ON THE SKOROKHOD REPRESENTATION THEOREM Jean Cortissoz In this paper we present a variant of the well known Skorokhod Representation Theorem. In our main result, given $S$ a Polish space, to a given continous path $\alpha$ in the space of probability measures on $S$, we associate a continuous path in the space of $S$-valued random variables on a nonatomic probability space (endowed with the topology of the convergence in probability). We call this associated path a lifting of $\alpha$. an interesting feature of our result is that we can fix the endpoints ("boundary values") of the lifting of $\alpha$, as long as their distribution correspond to the endpoints ("boundary values") of $\alpha$. We also discuss an $n$-dimensional generalization of this result. http://front.math.ucdavis.edu/math.PR/0601524 --------------------------------------------------------------- 4102. CORRECTING NEWTON-C\^{O}TES INTEGRALS BY L\'{E}VY AREAS Ivan Nourdin (PMA) and Thomas Simon (DP) In this note we introduce the notion of Newton-C\^{o}tes integral corrected by L\'{e}vy areas, which enables us to consider integrals of the type $\int f(y) dx,$ where f is a $C^{2m}$ function and $x, y$ are real H\"{o} lderian functions with index > 1/(2m+1), for any integer m. We show that this concept extends the Newton-C\^{o}tes integral introduced in (Gradinaru et al., Ann. Inst. H. Poincar\'{e} Probab. Statist. 41 (4), 781-806, 2005), to a larger class of integrands. Then, we give a theorem of existence and uniqueness for differential equations driven by x, interpreted using this new integral. http://front.math.ucdavis.edu/math.PR/0601544 --------------------------------------------------------------- 4103. UNIFORM IN BANDWIDTH CONSISTENCY OF LOCAL POLYNOMIAL REGRESSION FUNCTION ESTIMATORS Julia Dony and Uwe Einmahl and David M. Mason We generalize a method for proving uniform in bandwidth consistency results for kernel type estimators developed by the two last named authors. Such results are shown to be useful in establishing consistency of local polynomial estimators of the regression function. http://front.math.ucdavis.edu/math.ST/0601548 --------------------------------------------------------------- 4104. AN ADAPTIVE EULER-MARUYAMA SCHEME FOR SDES: CONVERGENCE AND STABILITY H. Lamba and J.C. Mattingly and A.M. Stuart The understanding of adaptive algorithms for SDEs is an open area where many issues related to both convergence and stability (long time behaviour) of algorithms are unresolved. This paper considers a very simple adaptive algorithm, based on controlling only the drift component of a time- step. Both convergence and stability are studied. The primary issue in the convergence analysis is that the adaptive method does not necessarily drive the time-steps to zero with the user-input tolerance. This possibility must be quantified and shown to have low probability. The primary issue in the stability analysis is ergodicity. It is assumed that the noise is non-degenerate, so that the diffusion process is elliptic, and the drift is assumed to satisfy a coercivity condition. The SDE is then geometrically ergodic (converges to statistical equilibrium exponentially quickly). If the drift is not linearly bounded then explicit fixed time-step approximations, such as the Euler-Maruyama scheme, may fail to be ergodic. In this work, it is shown that the simple adaptive time-stepping strategy cures this problem. In addition to proving ergodicity, an exponential moment bound is also proved, generalizing a result known to hold for the SDE itself. http://front.math.ucdavis.edu/math.NA/0601029 --------------------------------------------------------------- 4105. CONVEXITY PRESERVING JUMP-DIFFUSION MODELS FOR OPTION PRICING Erik Ekstr\"om and Johan Tysk We investigate which jump-diffusion models are convexity preserving. The study of convexity preserving models is motivated by monotonicity results for such models in the volatility and in the jump parameters. We give a necessary condition for convexity to be preserved in several-dimensional jump- diffusion models. This necessary condition is then used to show that, within a large class of possible models, the only convexity preserving models are the ones with linear coefficients. http://front.math.ucdavis.edu/math.AP/0601526 --------------------------------------------------------------- 4106. NUMBER OF COMPLETE N-ARY SUBTREES ON GALTON-WATSON FAMILY TREES George P. Yanev and Ljuben Mutafchiev We associate with a Bienayme-Galton-Watson branching process a family tree rooted at the ancestor. For a positive integer N, define a complete N- ary tree to be the family tree of a deterministic branching process with offspring generating function s^N. We study the random variables V(N,n) and V (N) counting the number of disjoint complete N-ary subtrees, rooted at the ancestor, and having height n and infinity, respectively. Dekking (1991) and Pakes and Dekking (1991) find recursive relations for Pr(V(N,n)>0) and Pr(V(N)>0) involving the offspring probability generation function (pgf) and its derivatives. We extend their results determining the probability distributions of V(N,n) and V(N). It turns out that they can be expressed in terms of the offspring pgf, its derivatives, and the above probabilities. We show how the general results simplify in case of fractional linear, geometric, Poisson, and one-or-many offspring laws. http://front.math.ucdavis.edu/math.PR/0601585 --------------------------------------------------------------- 4107. MONTE CARLO ALGORITHM FOR LEAST DEPENDENT NON-NEGATIVE MIXTURE DECOMPOSITION Sergey A. Astakhov and Harald St\"ogbauer and Alexander Kraskov and Peter Grassberger We propose a simulated annealing algorithm (called SNICA for "stochastic non-negative independent component analysis") for blind decomposition of linear mixtures of non-negative sources with non-negative coefficients. The de-mixing is based on a Metropolis type Monte Carlo search for least dependent components, with the mutual information between recovered components as a cost function and their non-negativity as a hard constraint. Elementary moves are shears in two-dimensional subspaces and rotations in three-dimensional subspaces. The algorithm is geared at decomposing signals whose probability densities peak at zero, the case typical in analytical spectroscopy and multivariate curve resolution. The decomposition performance on large samples of synthetic mixtures and experimental data is much better than that of traditional blind source separation methods based on principal component analysis (MILCA, FastICA, RADICAL) and chemometrics techniques (SIMPLISMA, ALS, BTEM) The source codes of SNICA, MILCA and the MI estimator are freely available online at http://www.fz-juelich.de/nic/cs/software http://front.math.ucdavis.edu/physics/0601161 --------------------------------------------------------------- 4108. THE HYPERGROUP PROPERTY AND REPRESENTATION OF MARKOV KERNELS Dominique Bakry (LSProba) and Nolwen Huet (LSProba) In a number of situations, Markov operators appear to be a wonderful tool to provide useful information on measured spaces. In this article, we introduce the so-called hypergroup property for an orthonormal basis $(f\_n)$ so as to describe all Markov operators which have the $f\_n$ as eigenvectors. In the finite case, this property appears as the dual of the GKS property linked with correlation inequalities in statistical mechanics. The representation theory of groups provide generic examples where these two properties are verified, although this group structure is not necessary in general. The hypergroup property also holds for Sturm-Liouville bases associated with log- concave symmetric measure on a compact interval, as stated in Achour's theorem. We relax this symmetry condition in view of extensions in Riemannian geometry for manifolds with non negative Ricci curvature. In the case of Jacobi polynomials with non-symmetric parameters, we need Gasper's theorem. The proof we present is based on a natural interpretation of these polynomials as harmonic functions, and gives a representation of them as the moments of a complex variable. http://front.math.ucdavis.edu/math.PR/0601605 --------------------------------------------------------------- 4109. COAGULATION FRAGMENTATION LAWS INDUCED BY GENERAL COAGULATIONS OF TWO-PARAMETER POISSON-DIRICHLET PROCESSES Man-Wai Ho and Lancelot F. James and John W. Lau Pitman~(1999) describes a duality relationship between fragmentation and coagulation operators. An explicit relationship is described for the two-parameter Poisson-Dirichlet laws, with parameters {\footnotesize $(\alpha,\theta)$} and $(\beta,\theta/\alpha)$, wherein $PD(\alpha, \theta)$ is coagulated by $PD(\beta,\theta/\alpha)$ for $0<\alpha<1$, $0 \leq \beta<1$ and $-\beta<\theta/\alpha$. This remarkable explicit agreement was obtained by combinatorial methods via exchangeable partition probability functions~(EPPF). This work discusses an alternative analysis which can feasibly extend the characterizations above to more general models of $PD(\alpha,\theta)$ coagulated with some law $Q$. The analysis exploits distributional relationships between compositions of species sampling random probability measures and coagulation operators and recent work on Cauchy-Stieltjes transforms of random probability measures by Vershik, Yor and Tsilevich (2004) and James (2002). We use this to obtain explicit descriptions in the case where {\footnotesize $Q$} corresponds to a large class of power tempered Poisson Kingman models analyzed in James~(2002). That is, explicit results are obtained for models outside of the $PD(\beta,\theta/\alpha)$ family. http://front.math.ucdavis.edu/math.PR/0601608 From pas at www.economia.unimi.it Fri May 5 09:50:39 2006 From: pas at www.economia.unimi.it (pas@www.economia.unimi.it) Date: Fri May 5 09:51:03 2006 Subject: [Pas] Probability Abstracts 92 Message-ID: <79B2AA22-CE07-4D9E-95AD-EAD48829EC43@unimi.it> May 5, 2006 Letter 92 Probability Abstract Service Abstracts from Mar-1-2006 to Apr-28-2006 --------------------------------------------------------------- 4110. LARGE DEVIATION FOR DIFFUSIONS AND HAMILTON--JACOBI EQUATION IN HILBERT SPACES Jin Feng Large deviation for Markov processes can be studied by Hamilton--Jacobi equation techniques. The method of proof involves three steps: First, we apply a nonlinear transform to generators of the Markov processes, and verify that limit of the transformed generators exists. Such limit induces a Hamilton--Jacobi equation. Second, we show that a strong form of uniqueness (the comparison principle) holds for the limit equation. Finally, we verify an exponential compact containment estimate. The large deviation principle then follows from the above three verifications. This paper illustrates such a method applied to a class of Hilbert-space-valued small diffusion processes. The examples include stochastically perturbed Allen--Cahn, Cahn-- Hilliard PDEs and a one-dimensional quasilinear PDE with a viscosity term. We prove the comparison principle using a variant of the Tataru method. We also discuss different notions of viscosity solution in infinite dimensions in such context. http://front.math.ucdavis.edu/math.PR/0602655 --------------------------------------------------------------- 4111. PASSAGE OF L\'{E}VY PROCESSES ACROSS POWER LAW BOUNDARIES AT SMALL TIMES Jean Bertoin (PMA) and Ronald A. Doney and Ross A. Maller (CMA) We wish to characterise when a L\'{e}vy process $X\_t$ crosses boundaries like $t^\kappa$, $\kappa>0$, in a one or two-sided sense, for small times $t$; thus, we enquire when $\limsup\_{t\downarrow 0}|X\_t|/t^{\kappa}$, $\limsup\_{t\downarrow 0}X\_t/t^{\kappa}$ and/or $\liminf\_{t\downarrow 0}X\_t/t^{\kappa}$ are almost surely (a.s.) finite or infinite. Necessary and sufficient conditions are given for these possibilities for all values of $\kappa>0$. Often (for many values of $\kappa$), when the limsups are finite a.s., they are in fact zero, as we show, but the limsups may in some circumstances take finite, nonzero, values, a.s. In general, the process crosses one or two-sided boundaries in quite different ways, but surprisingly this is not so for the case $\kappa=1/2$. An integral test is given to distinguish the possibilities in that case. Some results relating to other norming sequences for $X$, and when $X$ is centered at a nonstochastic function, are also given. http://front.math.ucdavis.edu/math.PR/0603274 --------------------------------------------------------------- 4112. CONSTRUCTIVE NO-ARBITRAGE CRITERION UNDER TRANSACTION COSTS IN THE CASE OF FINITE DISCRETE TIME Dmitry B. Rokhlin We obtain a constructive criterion for robust no-arbitrage in discrete-time market models with transaction costs. This criterion is expressed in terms of the supports of the regular conditional upper distributions of the solvency cones. We also consider the model with a bank account. A method for construction of arbitrage strategies is proposed. http://front.math.ucdavis.edu/math.PR/0603284 --------------------------------------------------------------- 4113. INDUCED GELATION IN A TWO-SITE SPATIAL COAGULATION MODEL Rainer Siegmund-Schultze and Wolfgang Wagner A two-site spatial coagulation model is considered. Particles of masses m and n at the same site form a new particle of mass m+n at rate mn. Independently, particles jump to the other site at a constant rate. The limit (for increasing particle numbers) of this model is expected to be non-deterministic after the gelation time, namely, one or two giant particles randomly jump between the two sites. Moreover, a new effect of induced gelation is observed - the gelation happening at the site with the larger initial number of monomers immediately induces gelation at the other site. Induced gelation is shown to be of logarithmic order. The limiting behaviour of the model is derived rigorously up to the gelation time, while the expected post-gelation behaviour is illustrated by a numerical simulation. http://front.math.ucdavis.edu/math.PR/0603300 --------------------------------------------------------------- 4114. CUBE ROOT FLUCTUATIONS FOR THE CORNER GROWTH MODEL ASSOCIATED TO THE EXCLUSION PROCESS Marton Balazs and Eric Cator and Timo Seppalainen We study the last-passage growth model on the planar integer lattice with exponential weights. With boundary conditions that represent the equilibrium exclusion process as seen from a particle right after its jump we prove that the variance of the last-passage time in a characteristic direction is of order t^{2/3}. With more general boundary conditions that include the rarefaction fan case we show that the last-passage time fluctuations are still of order t^{1/3}, and also that the transversal fluctuations of the maximal path have order t^{2/3}. We adapt and then build on a recent study of Hammersley's process by Cator and Groeneboom, and also utilize the competition interface introduced by Ferrari, Martin and Pimentel. The arguments are entirely probabilistic, and no use is made of the combinatorics of Young tableaux or methods of asymptotic analysis. http://front.math.ucdavis.edu/math.PR/0603306 --------------------------------------------------------------- 4115. WEAK DISORDER IN FIBONACCI SEQUENCES E. Ben-Naim and P.L. Krapivsky We study how weak disorder affects the growth of the Fibonacci series. We introduce a family of stochastic sequences that grow by the normal Fibonacci recursion with probability 1-epsilon, but follow a different recursion rule with a small probability epsilon. We focus on the weak disorder limit and obtain the Lyapunov exponent, that characterizes the typical growth of the sequence elements, using perturbation theory. The limiting distribution for the ratio of consecutive sequence elements is obtained as well. A number of variations to the basic Fibonacci recursion including shift, doubling, and copying are considered. http://front.math.ucdavis.edu/cond-mat/0603117 --------------------------------------------------------------- 4116. UNIVERSALITY FOR MATHEMATICAL AND PHYSICAL SYSTEMS Percy Deift All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all governed by the same laws and formulae of thermodynamics. In this paper we describe some recent history of universality ideas in physics starting with Wigner's model for the scattering of neutrons off large nuclei and show how these ideas have led mathematicians to investigate universal behavior for a variety of mathematical systems. This is true not only for systems which have a physical origin, but also for systems which arise in a purely mathematical context such as the Riemann hypothesis, and a version of the card game solitaire called patience sorting. http://front.math.ucdavis.edu/math-ph/0603038 --------------------------------------------------------------- 4117. ON FINITE-DIMENSIONAL PROJECTIONS OF DISTRIBUTIONS FOR SOLUTIONS OF RANDOMLY FORCED PDE'S Andrei Agrachev (SISSA-Isas) and Sergei Kuksin (Mathematics Department of Heriot-Watt University), Andrey Sarychev (DMD), Armen Shirikyan (LM-Orsay) The paper is devoted to studying the image of probability measures on a Hilbert space under finite-dimensional analytic maps. We establish sufficient conditions under which the image of a measure has a density with respect to the Lebesgue measure and continuously depends on the map. The results obtained are applied to the 2D Navier--Stokes equations perturbed by various random forces of low dimension. http://front.math.ucdavis.edu/math.AP/0603295 --------------------------------------------------------------- 4118. SIMULATION OF DISCRETE SYSTEMS USING PROBABILISTIC SEQUENTIAL SYSTEMS Maria A. Avino-Diaz and Gabriela Bulancea and Oscar Moreno In this paper we introduce the idea of probability in the definition of a Sequential Dynamical System (SDS), thus obtaining a new concept, that of Probabilistic Sequential System (PSS). Due to its particular dynamic, the Probabilistic Boolean Network (PBN) model has been applied to genetic regulatory networks. The model we introduce combines the sequential aspect of the SDSs and the dynamic of the PBNs. The notion of simulation of a PSS is introduced using the concept of morphism of PSSs. We prove that the PSSs with the PSS-morphisms form a category PSS. Several examples of morphisms, subsystems and simulations are given. http://front.math.ucdavis.edu/math.DS/0603289 --------------------------------------------------------------- 4119. SPECIAL HOMOMORPHISMS BETWEEN PROBABILISTIC GENE REGULATORY NETWORKS Maria A. Avino-Diaz In this paper we study finite dynamical systems with $n$ functions acting on the same set $X$, and probabilities assigned to these functions, that it is called Probabilistic Regulatory Gene Networks (PRN. his concept is the same or a natural generalization of the concept Probabilistic Boolean Networks (PBN), introduced by I. Shmulevich, E. Dougherty, and W. Zhang, particularly the model PBN has been using to describe genetic networks and has therapeutic applications. In PRNs the most important question is to describe the steady states of the systems, so in this paper we pay attention to the idea of transforming a network to another without lost all the properties, in particular the probability distribution. Following this objective we develop the concepts of homomorphism and $\epsilon$-homomorphism of probabilistic regulatory networks, since these concepts bring the properties from one networks to another. Projections are special homomorphisms, and they always induce invariant subnetworks that contain all cycles and steady states in the network. http://front.math.ucdavis.edu/math.DS/0603291 --------------------------------------------------------------- 4120. PROBABILISTIC GENE REGULATORY NETWORKS, ISOMORPHISMS OF MARKOV CHAINS Maria A. Avino-Diaz In this paper we study homomorphisms of Probabilistic Regulatory Gene Networks(PRN) introduced in arXiv:math.DS/0603289 v1 13 Mar 2006. The model PRN is a natural generalization of the Probabilistic Boolean Networks (PBN), introduced by I. Shmulevich, E. Dougherty, and W. Zhang in 2001, that has been using to describe genetic networks and has therapeutic applications. In this paper, our main objectives are to apply the concept of homomorphism and $\epsilon$-homomorphism of probabilistic regulatory networks to the dynamic of the networks. The meaning of $\epsilon$ is that these homomorphic networks have similar distributions and the distance between the distributions is upper bounded by $\epsilon$. Additionally, we prove that the class of PRN together with the homomorphisms form a category with products and coproducts. Projections are special homomorphisms, and they always induce invariant subnetworks that contain all the cycles and steady states in the network. Here, it is proved that the $\epsilon$-homomorphism for $0<\epsilon<1$ produce simultaneous Markov Chains in both networks, that permit to introduce the concept of $\epsilon$-isomorphism of Markov Chains, and similar networks. http://front.math.ucdavis.edu/math.DS/0603302 --------------------------------------------------------------- 4121. STATE DEPENDENT UTILITY Jaime A. Londo\~no We propose a new approach to utilities that is consistent with state-dependent utilities. In our model utilities reflect the level of consumption satisfaction of flows of cash in future times as they are valued when the economic agents are making their consumption and investment decisions. The theoretical framework used for the model is one proposed by the author in Dynamic State Tameness {arXiv:math.PR/0509139}. The proposed framework is a generalization of the theory of Brownian flows and can be applied to those processes that are the solutions of classical It^o stochastic differential equations, even when the volatilities and drifts are just locally $\delta$-Holder continuous for some $\delta>0$. We develop the martingale methodology for the solution of the problem of optimal consumption and investment. Complete solutions of the optimal consumption and portfolio problem are obtained in a very general setting which includes several functional forms for utilities in the current literature, and consider general restrictions on minimal wealths. As a secondary result we obtain a suitable representation for straightforward numerical computations of the optimal consumption and investment strategies. http://front.math.ucdavis.edu/math.PR/0603316 --------------------------------------------------------------- 4122. SYSTEMATIC SCAN FOR SAMPLING COLORINGS Martin Dyer and Leslie Ann Goldberg and Mark Jerrum We address the problem of sampling colorings of a graph $G$ by Markov chain simulation. For most of the article we restrict attention to proper $q$-colorings of a path on $n$ vertices (in statistical physics terms, the one-dimensional $q$-state Potts model at zero temperature), though in later sections we widen our scope to general ``$H$-colorings'' of arbitrary graphs $G$. Existing theoretical analyses of the mixing time of such simulations relate mainly to a dynamics in which a random vertex is selected for updating at each step. However, experimental work is often carried out using systematic strategies that cycle through coordinates in a deterministic manner, a dynamics sometimes known as systematic scan. The mixing time of systematic scan seems more difficult to analyze than that of random updates, and little is currently known. In this article we go some way toward correcting this imbalance. By adapting a variety of techniques, we derive upper and lower bounds (often tight) on the mixing time of systematic scan. An unusual feature of systematic scan as far as the analysis is concerned is that it fails to be time reversible. http://front.math.ucdavis.edu/math.PR/0603323 --------------------------------------------------------------- 4123. PATTERN DENSITIES IN FLUID DIMER MODELS Cedric Boutillier In this paper, we introduce a family of observables for the dimer model on a bi-periodic bipartite planar graph, called pattern density fields. We study the scaling limit of these objects for liquid and gaseous Gibbs measures of the dimer model, and prove that they converge to a linear combination of a derivative of the Gaussian massless free field and an independent white noise. http://front.math.ucdavis.edu/math.PR/0603324 --------------------------------------------------------------- 4124. MEAN FIELD CONVERGENCE OF A MODEL OF MULTIPLE TCP CONNECTIONS THROUGH A BUFFER IMPLEMENTING RED D. R. McDonald and J. Reynier RED (Random Early Detection) has been suggested when multiple TCP sessions are multiplexed through a bottleneck buffer. The idea is to detect congestion before the buffer overflows by dropping or marking packets with a probability that increases with the queue length. The objectives are reduced packet loss, higher throughput, reduced delay and reduced delay variation achieved through an equitable distribution of packet loss and reduced synchronization. Baccelli, McDonald and Reynier [Performance Evaluation 11 (2002) 77--97] have proposed a fluid model for multiple TCP connections in the congestion avoidance regime multiplexed through a bottleneck buffer implementing RED. The window sizes of each TCP session evolve like independent dynamical systems coupled by the queue length at the buffer. The key idea in [Performance Evaluation 11 (2002) 77--97] is to consider the histogram of window sizes as a random measure coupled with the queue. Here we prove the conjecture made in [Performance Evaluation 11 (2002) 77--97] that, as the number of connections tends to infinity, this system converges to a deterministic mean-field limit comprising the window size density coupled with a deterministic queue. http://front.math.ucdavis.edu/math.PR/0603325 --------------------------------------------------------------- 4125. LARGE DEVIATION ASYMPTOTICS AND CONTROL VARIATES FOR SIMULATING LARGE FUNCTIONS Sean P. Meyn Consider the normalized partial sums of a real-valued function $F$ of a Markov chain, \[\phi_n:=n^{-1}\sum_{k=0}^{n-1}F(\Phi(k)),\qquad n\ge1. \] The chain $\{\Phi(k):k\ge0\}$ takes values in a general state space $ \mathsf {X}$, with transition kernel $P$, and it is assumed that the Lyapunov drift condition holds: $PV\le V-W+b\mathbb{I}_C$ where $V:\mathsf {X}\to(0,\infty)$, $W:\mathsf {X}\to[1,\infty)$, the set $C$ is small and $W$ dominates $F$. Under these assumptions, the following conclusions are obtained: 1. It is known that this drift condition is equivalent to the existence of a unique invariant distribution $\pi$ satisfying $\pi(W)<\infty$, and the law of large numbers holds for any function $F$ dominated by $W$: \[\phi_n\to\phi:=\pi(F),\qquad{a.s.}, n\to\infty.\] 2. The lower error probability defined by $\mathsf {P}\{\phi_n\le c\}$, for $c<\phi$, $n \ge1$, satisfies a large deviation limit theorem when the function $F$ satisfies a monotonicity condition. Under additional minor conditions an exact large deviations expansion is obtained. 3. If $W$ is near-monotone, then control-variates are constructed based on the Lyapunov function $V$, providing a pair of estimators that together satisfy nontrivial large asymptotics for the lower and upper error probabilities. In an application to simulation of queues it is shown that exact large deviation asymptotics are possible even when the estimator does not satisfy a central limit theorem. http://front.math.ucdavis.edu/math.PR/0603328 --------------------------------------------------------------- 4126. CORRECTION. IMPROPER REGULAR CONDITIONAL DISTRIBUTIONS Teddy Seidenfeld and Mark J. Schervish and Joseph B. Kadane Correction to Annals of Probability 29 (2001) 1612--1624 [doi:10.1214/aop/1015345764]. http://front.math.ucdavis.edu/math.PR/0603012 --------------------------------------------------------------- 4127. ASYMPTOTIC THEOREMS OF SEQUENTIAL ESTIMATION-ADJUSTED URN MODELS Li-X. Zhang and Feifang Hu and Siu Hung Cheung The Generalized P\'{o}lya Urn (GPU) is a popular urn model which is widely used in many disciplines. In particular, it is extensively used in treatment allocation schemes in clinical trials. In this paper, we propose a sequential estimation-adjusted urn model (a nonhomogeneous GPU) which has a wide spectrum of applications. Because the proposed urn model depends on sequential estimations of unknown parameters, the derivation of asymptotic properties is mathematically intricate and the corresponding results are unavailable in the literature. We overcome these hurdles and establish the strong consistency and asymptotic normality for both the patient allocation and the estimators of unknown parameters, under some widely satisfied conditions. These properties are important for statistical inferences and they are also useful for the understanding of the urn limiting process. A superior feature of our proposed model is its capability to yield limiting treatment proportions according to any desired allocation target. The applicability of our model is illustrated with a number of examples. http://front.math.ucdavis.edu/math.PR/0603329 --------------------------------------------------------------- 4128. ON THE ASYMPTOTICS OF THE SUPREMUM OF A RANDOM WALK: THE PRINCIPLE OF A SINGLE BIG JUMP IN THE LIGHT-TAILED CASE Stan Zachary and Serguei Foss We study the distribution of the maximum $M$ of a random walk whose increments have a distribution with negative mean and belonging, for some $\gamma\ge0$, to the class $\mathcal{S}_{\gamma}$ introduced by Chover, Ney, and Weinger (1973). For $\gamma>0$, we give a probabilistic derivation of the asymptotic tail distribution of $M$ and show that, as in the case $ \gamma=0$, extreme values of $M$ are in general attained through some single large increment in the random walk. http://front.math.ucdavis.edu/math.PR/0603330 --------------------------------------------------------------- 4129. INDIVIDUAL VERSUS CLUSTER RECOVERIES WITHIN A SPATIALLY STRUCTURED POPULATION L. Belhadji and N. Lanchier Stochastic modeling of disease dynamics has had a long tradition. Among the first epidemic models including a spatial structure in the form of local interactions is the contact process. In this article we investigate two extensions of the contact process describing the course of a single disease within a spatially structured human population distributed in social clusters. That is, each site of the $d$-dimensional integer lattice is occupied by a cluster of individuals; each individual can be healthy or infected. The evolution of the disease depends on three parameters, namely the outside infection rate which models the interactions between the clusters, the within infection rate which takes into account the repeated contacts between individuals in the same cluster, and the size of each social cluster. For the first model, we assume cluster recoveries, while individual recoveries are assumed for the second one. The aim is to investigate the existence of nontrivial stationary distributions for both processes depending on the value of each of the three parameters. Our results show that the probability of an epidemic strongly depends on the recovery mechanism. http://front.math.ucdavis.edu/math.PR/0603331 --------------------------------------------------------------- 4130. A ZERO-ONE LAW FOR FIRST-ORDER LOGIC ON RANDOM IMAGES David Coupier (MAP5) and Agn\`{e}s Desolneux (MAP5) and Bernard Ycart (LMC - IMAG) For an $n\times n$ random image with independent pixels, black with probability $p(n)$ and white with probability $1-p(n)$, the probability of satisfying any given first-order sentence tends to 0 or 1, provided both $p(n)n^{\frac{2}{k}}$ and $(1-p(n))n^{\frac{2}{k}}$ tend to 0 or $+ \infty$, for any integer $k$. The result is proved by computing the threshold function for basic local sentences, and applying Gaifman's theorem. http://front.math.ucdavis.edu/math.PR/0603333 --------------------------------------------------------------- 4131. SOME STRONG LIMIT THEOREMS FOR THE LARGEST ENTRIES OF SAMPLE CORRELATION MATRICES Deli Li and Andrew Rosalsky Let $\{X_{k,i};i\geq 1,k\geq 1\}$ be an array of i.i.d. random variables and let $\{p_n;n\geq 1\}$ be a sequence of positive integers such that $n/ p_n$ is bounded away from 0 and $\infty$. For $W_n=\max_{1\leq i1/2)$, (ii) $\lim_{n\to \infty}n^{1-\alpha}L_n=0$ a.s. $(1/2<\alpha \leq 1)$, (iii) $\lim_{n\to \infty}\frac{W_n}{\sqrt{n\log n}}=2$ a.s. and (iv) $\lim_{n\to \infty}(\frac{n}{\log n})^{1/2}L_n=2$ a.s. are shown to hold under optimal sets of conditions. These results follow from some general theorems proved for arrays of i.i.d. two-dimensional random vectors. The converses of the limit laws (i) and (iii) are also established. The current work was inspired by Jiang's study of the asymptotic behavior of the largest entries of sample correlation matrices. http://front.math.ucdavis.edu/math.PR/0603334 --------------------------------------------------------------- 4132. STOCHASTIC SPATIAL MODELS OF HOST-PATHOGEN AND HOST-MUTUALIST INTERACTIONS I N. Lanchier and C. Neuhauser Mutualists and pathogens, collectively called symbionts, are ubiquitous in plant communities. While some symbionts are highly host-specific, others associate with multiple hosts. The outcomes of multispecies host- symbiont interactions with different degrees of specificity are difficult to predict at this point due to a lack of a general conceptual framework. Complicating our predictive power is the fact that plant populations are spatially explicit, and we know from past research that explicit space can profoundly alter plant-plant interactions. We introduce a spatially explicit, stochastic model to investigate the role of explicit space and host-specificity in multispecies host-symbiont interactions. We find that in our model, pathogens can significantly alter the spatial structure of plant communities, promoting coexistence, whereas mutualists appear to have only a limited effect. Effects are more pronounced the more host-specific symbionts are. http://front.math.ucdavis.edu/math.PR/0603335 --------------------------------------------------------------- 4133. IMAGE DENOISING BY STATISTICAL AREA THRESHOLDING David Coupier (MAP5) and Agn\`{e}s Desolneux (MAP5) and Bernard Ycart (LMC - IMAG) Area openings and closings are morphological filters which efficiently suppress impulse noise from an image, by removing small connected components of level sets. The problem of an objective choice of threshold for the area remains open. Here, a mathematical model for random images will be considered. Under this model, a Poisson approximation for the probability of appearance of any local pattern can be computed. In particular, the probability of observing a component with size larger than $k$ in pure impulse noise has an explicit form. This permits the definition of a statistical test on the significance of connected components, thus providing an explicit formula for the area threshold of the denoising filter, as a function of the impulse noise probability parameter. Finally, using threshold decomposition, a denoising algorithm for grey level images is proposed. http://front.math.ucdavis.edu/math.PR/0603337 --------------------------------------------------------------- 4134. THE ARCSINE LAW AS A UNIVERSAL AGING SCHEME FOR TRAP MODELS Gerard Ben Arous and Jiri Cerny We give a general proof of aging for trap models using the arcsine law for stable subordinators. This proof is based on abstract conditions on the potential theory of the underlying graph and on the randomness of the trapping landscape. We apply this proof to aging for trap models on large two-dimensional tori and for trap dynamics of the Random Energy Model on a broad range of time scales. http://front.math.ucdavis.edu/math.PR/0603340 --------------------------------------------------------------- 4135. DISCRETE IT\^O FORMULAS AND THEIR APPLICATIONS TO STOCHASTIC NUMERICS Jir\^o Akahori This is a survey note of the author's observations on the discrete-time analogues of It\^o formulas. http://front.math.ucdavis.edu/math.PR/0603341 --------------------------------------------------------------- 4136. DYNAMICS OF TRAP MODELS Gerard Ben Arous and Jiri Cerny These notes cover one of the topics of the class given in the Les Houches Summer School ``Mathematical statistical physics'' in July 2005. The lectures tried to give a summary of the recent mathematical results about the long-time behaviour of dynamics of (mean-field) spin-glasses and other disordered media. We have chosen here to restrict the scope of these notes to the dynamics of trap models only, but to cover this topic in somewhat more depth. http://front.math.ucdavis.edu/math.PR/0603344 --------------------------------------------------------------- 4137. CORRECTION. CENTRAL LIMIT THEOREMS FOR ADDITIVE FUNCTIONALS OF THE SIMPLE EXCLUSION PROCESS S. Sethuraman Correction to Annals of Probability 28 (2000) 277--302 [doi:10.1214/aop/1019160120]. http://front.math.ucdavis.edu/math.PR/0603014 --------------------------------------------------------------- 4138. SECOND CLASS PARTICLES AND CUBE ROOT ASYMPTOTICS FOR HAMMERSLEY'S PROCESS Eric Cator and Piet Groeneboom We show that, for a stationary version of Hammersley's process, with Poisson sources on the positive x-axis and Poisson sinks on the positive y- axis, the variance of the length of a longest weakly North-East path $L(t,t)$ from $(0,0)$ to $(t,t)$ is equal to $2\E(t-X(t))_+$, where $X(t)$ is the location of a second class particle at time $t$. This implies that both $\E(t-X (t))_+$ and the variance of $L(t,t)$ are of order $t^{2/3}$. Proofs are based on the relation between the flux and the path of a second class particle, continuing the approach of Cator and Groeneboom (2005). http://front.math.ucdavis.edu/math.PR/0603345 --------------------------------------------------------------- 4139. RIGHT-PERMUTATIVE CELLULAR AUTOMATA ON TOPOLOGICAL MARKOV CHAINS Marcelo Sobottka In this paper we consider cellular automata $(\mathfrak{G},\Phi)$ with algebraic local rules and such that $\mathfrak{G}$ is a topological Markov chain which has a structure compatible to this local rule. We characterize such cellular automata and study the convergence of the Ces\`aro mean distribution of the iterates of any probability measure with complete connections and summable decay. http://front.math.ucdavis.edu/math.DS/0603326 --------------------------------------------------------------- 4140. A SUBDIFFUSIVE BEHAVIOUR OF RECURRENT RANDOM WALK IN RANDOM ENVIRONMENT ON A REGULAR TREE Yueyun Hu (LAGA) and Zhan Shi (PMA) We are interested in the random walk in random environment on an infinite tree. Lyons and Pemantle [11] give a precise recurrence/transience criterion. Our paper focuses on the almost sure asymptotic behaviours of a recurrent random walk $(X\_n)$ in random environment on a regular tree, which is closely related to Mandelbrot [13]'s multiplicative cascade. We prove, under some general assumptions upon the distribution of the environment, the existence of a new exponent $\nu\in (0, {1\over 2}]$ such that $\max\_{0\le i \le n} |X\_i|$ behaves asymptotically like $n^{\nu}$. The value of $\nu$ is explicitly formulated in terms of the distribution of the environment. http://front.math.ucdavis.edu/math.PR/0603363 --------------------------------------------------------------- 4141. BETA-PATHS IN THE HAMMERSLEY PROCESS Cristian Coletti and Leandro P. R. Pimentel We study the asymptotics of beta-paths in the Hammersley process with sources and sinks, in the rarefaction regime. We derive a strong law of large number for those paths and we show that its fluctuation exponent is at most 2/3. Examples of beta-paths are the space-time path of a second-class particle in the Hammersley process and also the space-time path of the interface between two PNG droplets. http://front.math.ucdavis.edu/math.PR/0603382 --------------------------------------------------------------- 4142. TESTING STATISTICAL HYPOTHESIS ON RANDOM TREES Jorge R. Busch and Pablo A. Ferrari and A. Georgina Flesia and Ricardo Fraiman and Sebastian Grynberg To distinguish between populations of trees, we consider the hypothesis test proposed recently by Balding, Ferrari, Fraiman and Sued (BFFS--test). A direct approach to calculate effectively the test statistic is quite difficult, since it is based on a supremum defined over the space of all trees, which grows exponentially fast. We show how to transform this problem into a max- flow over a network which can be solved using a Ford Fulkerson algorithm in polynomial time on the maximal number of vertices of the random tree. We also describe conditions that imply the characterization of the measure by the marginal distributions of each node of the random tree, which validate the use of the BFFS--test for measure discrimination. The performance of the test is studied via simulations on Galton-Watson processes. http://front.math.ucdavis.edu/math.ST/0603378 --------------------------------------------------------------- 4143. DEVIATION BOUNDS FOR ADDITIVE FUNCTIONALS OF MARKOV PROCESS Patrick Cattiaux (CMAP and Modal'x) and Arnaud Guillin (CEREMADE) In this paper we derive non asymptotic deviation bounds for $$\P_\nu (|\frac 1t \int_0^t V(X_s) ds - \int V d\mu | \geq R)$$ where $X$ is a $\mu$ stationary and ergodic Markov process and $V$ is some $\mu$ integrable function. These bounds are obtained under various moments assumptions for $V$, and various regularity assumptions for $\mu$. Regularity means here that $\mu$ may satisfy various functional inequalities (F- Sobolev, generalized Poincar\'e etc...). http://front.math.ucdavis.edu/math.PR/0603021 --------------------------------------------------------------- 4144. WEAK DISORDER FOR LOW DIMENSIONAL POLYMERS: THE MODEL OF STABLE LAWS Francis Comets (PMA) In this paper, we consider directed polymers in random environment with long range jumps in discrete space and time. We extend to this case some techniques, results and classifications known in the usual short range case. However, some properties are drastically different when the underlying random walk belongs to the domain of attraction of an $\a$-stable law. For instance, we construct natural examples of directed polymers in random environment which experience weak disorder in low dimension. http://front.math.ucdavis.edu/math.PR/0603390 --------------------------------------------------------------- 4145. TRANSIENT RANDOM WALKS ON A STRIP IN A RANDOM ENVIRONMENT Alexander Roitershtein We consider transient random walks on a strip in a random environment. The model was introduced by Bolthausen and Goldsheid in [4]. We derive a strong law of large numbers for the random walks in a general ergodic setup and obtain an annealed central limit theorem in the case of uniformly mixing environments. In addition, we prove that the law of the ``environment viewed from the position of the walker'' converges to a limiting distribution if the environment is an i.i.d. sequence. http://front.math.ucdavis.edu/math.PR/0603392 --------------------------------------------------------------- 4146. PROCESS LEVEL MODERATE DEVIATIONS FOR STABILIZING FUNCTIONALS Peter Eichelsbacher and Tomasz Schreiber Functionals of spatial point process often satisfy a weak spatial dependence condition known as stabilization. In this paper we prove process level moderate deviation principles (MDP) for such functionals, which are a level-3 result for empirical point fields as well as a level-2 result for empirical point measures. The level-3 rate function coincides with the so-called specific information. We show that the general result can be applied to prove MDPs for various particular functionals, including random sequential packing, birth-growth models, germ-grain models and nearest neighbor graphs. http://front.math.ucdavis.edu/math.PR/0603402 --------------------------------------------------------------- 4147. SOME SCALING LIMITS FOR A BROWNIAN POLYMER IN A GAUSSIAN MEDIUM Sergio De Carvalho Bezerra (IECN) and Samy Tindel (IECN) and Frederi Viens This paper provides information about the asymptotic behavior of a one-dimensional Brownian polymer in random medium represented by a space-time Gaussian field W assumed to be white noise in time and function- valued in space. According to the behavior of the spatial covariance W, we give sharp upper and lower bounds on the partition function's exponential rate (Lyapunov exponent), and on the growth (wandering exponent) of the polymer itself when the time parameter goes to infinity. http://front.math.ucdavis.edu/math.PR/0603404 --------------------------------------------------------------- 4148. LARGE DEVIATIONS FOR PAST-DEPENDENT RECURSIONS F. Klebaner and R. Liptser The Large Deviation Principle is established for stochastic models defined by past-dependent non linear recursions with small noise. In the Markov case we use the result to obtain an explicit expression for the asymptotics of exit time. http://front.math.ucdavis.edu/math.PR/0603407 --------------------------------------------------------------- 4149. PERMUTATIONS WITHOUT LONG DECREASING SUBSEQUENCES AND RANDOM MATRICES Piotr Sniady We study the shape of the Young diagram \lambda associated via the Robinson-Schensted-Knuth algorithm to a random permutation in S_n such that the length of the longest decreasing subsequence is not bigger than a fixed number d; in other words we study the restriction of the Plancherel measure to Young diagrams with at most d rows. We prove that in the limit n\to\infty the rows of \lambda behave like the eigenvalues of a certain random matrix (traceless Gaussian Unitary Ensemble) with d rows and columns. In particular, the length of the longest increasing subsequence of such a random permutation behaves asymptotically like the largest eigenvalue of the corresponding random matrix. http://front.math.ucdavis.edu/math.CO/0603401 --------------------------------------------------------------- 4150. WEAK SURVIVAL FOR BRANCHING RANDOM WALKS ON GRAPHS Daniela Bertacchi and Fabio Zucca We study weak and strong survival for branching random walks on multigraphs. We prove that, for a large class of multigraphs, weak survival is related to a geometrical parameter of the multigraph and that the existence of a pure weak phase is equivalent to nonamenability. Finally we study weak and strong critical behaviors of the branching random walk. http://front.math.ucdavis.edu/math.PR/0603412 --------------------------------------------------------------- 4151. MODERATE DEVIATIONS FOR SOME POINT MEASURES IN GEOMETRIC PROBABILITY Peter Eichelsbacher and Tomasz Schreiber and Joseph E. Yukich Functionals in geometric probability are often expressed as sums of bounded functions exhibiting exponential stabilization. Methods based on cumulant techniques and exponential modifications of measures show that such functionals satisfy moderate deviation principles. This leads to moderate deviation principles and laws of the iterated logarithm for random packing models as well as for statistics associated with germ-grain models and $k$ nearest neighbor graphs. http://front.math.ucdavis.edu/math.PR/0603022 --------------------------------------------------------------- 4152. CONVEX GEOMETRY OF MAX-STABLE DISTRIBUTIONS Ilya Molchanov It is shown that max-stable random vectors in $[0,\infty)^d$ with unit Fr\'echet marginals are in one to one correspondence with convex sets $K$ in $[0,\infty)^d$ called max-zonoids. The max-zonoids can be characterised as sets obtained as limits of Minkowski sums of simplices or, alternatively, as the selection expectation of a random simplex whose distribution is controlled by the spectral measure of the max-stable random vector. Furthermore, the cumulative distribution function $\Prob{\xi\leq x}$ of a max-stable random vector $\xi$ with unit Fr\'echet marginals is determined by the norm of the inverse to $x$, where all possible norms are given by the support functions of max-zonoids. As an application, geometrical interpretations of a number of well-known concepts from the theory of multivariate extreme values and copulas are provided. The convex geometry approach makes it possible to generalise a number of known results and to introduce new operations with max- stable random vectors. http://front.math.ucdavis.edu/math.PR/0603423 --------------------------------------------------------------- 4153. DIFFERENTIABILITY OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS IN HILBERT SPACES WITH MONOTONE GENERATORS Philippe Briand (IRMAR) and Fulvia Confortola The aim of the present paper is to study the regularity properties of the solution of a backward stochastic differential equation with a monotone generator in infinite dimension. We show some applications to the nonlinear Kolmogorov equation and to stochastic optimal control. http://front.math.ucdavis.edu/math.PR/0603428 --------------------------------------------------------------- 4154. ON THE INFERENCE OF SPARTAN SPATIAL RANDOM FIELD MODELS FOR GEOSTATISTICAL APPLICATIONS Samuel Elogne and Dionisis Hristopulos This paper focuses on the estimation of model parameters (model inference) for the class of Spartan Spatial Random Fields (SSRFs) introduced by Hristopulos (2003). The approach used for model inference involves calculation of sample constraints and fitting with respective ensemble constraints. The fitting leads to optimal SSRF parameters obtained by minimizing a suitable distance functional. We propose kernel-based estimators for calculating the sample constraints from data distributed on irregular sampling grids. We investigate the asymptotic properties of the estimators, and we establish a criterion for the selection of the kernel bandwidth parameters. The performance of the sample constraint estimators, as well as that of the SSRF inference procedure is evaluated by means of numerical simulations for different models of spatial dependence. http://front.math.ucdavis.edu/math.ST/0603430 --------------------------------------------------------------- 4155. CONVERGENCE OF APPROXIMATIONS OF MONOTONE GRADIENT SYSTEMS Lorenzo Zambotti We consider stochastic differential equations in a Hilbert space, perturbed by the gradient of a convex potential. We investigate the problem of convergence of a sequence of such processes. We propose applications of this method to reflecting O.U. processes in infinite dimension, to stochastic partial differential equations with reflection of Cahn-Hilliard type and to interface models. http://front.math.ucdavis.edu/math.PR/0603474 --------------------------------------------------------------- 4156. STATISTICAL PROPERTIES OF TOPOLOGICAL COLLET-ECKMANN MAPS Feliks Przytycki and Juan Rivera-Letelier We study geometric and statistical properties of complex rational maps satisfying the Topological Collet-Eckmann Condition. We show that every such a rational map possesses a unique conformal probability measure of minimal exponent, and that this measure is non-atomic, ergodic and that its Hausdorff dimension is equal to the Hausdorff dimension of the Julia set. Furthermore, we show that there is a unique invariant probability measure that is absolutely continuous with respect to this conformal measure, and we show that this measure is exponentially mixing (it has exponential decay of correlations) and that it satisfies the Central Limit Theorem. We also show that for a complex rational map f the existence of such an invariant measure characterizes the Topological Collet-Eckmann Condition, and that this measure is the unique equilibrium state with potential - HD (J(f)) ln |f'|. http://front.math.ucdavis.edu/math.DS/0603459 --------------------------------------------------------------- 4157. ON INEQUALITIES FOR SUMS OF BOUNDED RANDOM VARIABLES Iosif Pinelis Let $\eta_1,\eta_2,...$ be independent (but not necessarily identically distributed) zero-mean random variables (r.v.'s) such that $|\eta_i| \le1$ almost surely for all $i$, and let $Z$ stand for a standard normal r.v. Let $a_1,a_2,...$ be any real numbers such that $a_1^2+a_2^2+...=1.$ It is shown that then for all $x>0$ $$ \P(a_1\eta_1+a_2\eta_2+...\ge x) \le \P(Z\ge x-\la/x), $$ where $\la := \ln\frac{2e^3}9=1.495...$. The proof relies on (i) another probability inequality and (ii) a l'Hospital-type rule for monotonicity, both developed elsewhere. Extensions to (super) martingales are indicated. http://front.math.ucdavis.edu/math.PR/0603030 --------------------------------------------------------------- 4158. THEOREMS LIMIT WITH WEIGHT FOR THE VECTORIAL MARTINGALES TO CONTINUOUS TIME Faouzi Chaabane and Ahmed Kebaier We develop a general approach of the almost sure central limit theorem for the quasi-continuous vectorial martingales and we release a quadratic extension of this theorem while specifying speeds of convergence. As an application of this result we study the problem of estimate the variance of a process with stationary and idependent increments in statistics. http://front.math.ucdavis.edu/math.PR/0603492 --------------------------------------------------------------- 4159. EXPLICIT LAWS OF LARGE NUMBERS FOR RANDOM NEAREST-NEIGHBOUR TYPE GRAPHS Andrew R. Wade We give laws of large numbers (in the L^p sense) for the total length of the k-nearest neighbours (directed) graph and the j-th nearest neighbour (directed) graph in R^d, with power-weighted edges. We deduce a law of large numbers for the standard nearest neighbour (undirected) graph. We give the limiting constants, in the case of uniform random points in (0,1)^d, explicitly. Also, we give explicit laws of large numbers for the total power-weighted length of the Gabriel graph and two further graphs that are related to the standard nearest-neighbour graph: the on-line nearest-neighbour graph and the minimal directed spanning forest. http://front.math.ucdavis.edu/math.PR/0603559 --------------------------------------------------------------- 4160. LIMIT THEORY FOR THE RANDOM ON-LINE NEAREST-NEIGHBOUR GRAPH Mathew D. Penrose and Andrew R. Wade In the on-line nearest-neighbour graph (ONG), each point after the first in a sequence of points in R^d is joined by an edge to its nearest- neighbour amongst those points that precede it in the sequence. We study the large-sample asymptotic behaviour of the total power-weighted length of the ONG on uniform random points in (0,1)^d. In particular, for d=1 and weight exponent \alpha>1/2, the limiting distribution of the centred total weight is characterized by a distributional fixed-point equation. As an ancillary result, we give exact expressions for the expectation and variance of the standard nearest-neighbour (directed) graph on uniform random points in the unit interval. http://front.math.ucdavis.edu/math.PR/0603561 --------------------------------------------------------------- 4161. A GENERALIZATION OF THE CENTRAL LIMIT THEOREM CONSISTENT WITH NONEXTENSIVE STATISTICAL MECHANICS Sabir Umarov and Stanly Steinberg and Constantino Tsallis As well known, the standard central limit theorem plays a fundamental role in Boltzmann-Gibbs (BG) statistical mechanics. This important physical theory has been generalized by one of us (CT) in 1988 by using the entropy $S_q = \frac{1-\sum_i p_i^q}{q-1}$ (with $q \in \cal{R}$) instead of its particular case $S_1=S_{BG}= -\sum_i p_i \ln p_i$. The theory which emerges is usually referred to as {\it nonextensive statistical mechanics} and recovers the standard theory for $q=1$. During the last two decades, this $q$- generalized statistical mechanics has been successfully applied to a considerable amount of physically interesting complex phenomena. Conjectures and numerical indications available in the literature were since a few years suggesting the possibility of $q$-generalizations of the standard central limit theorem by allowing the random variables that are being summed to be correlated in some special manner, the case $q=1$ corresponding to standard probabilistic independence. This is precisely what we prove in the present paper for some range of $q$ which extends from below to above $q=1$. The attractor, in the usual sense of a central limit theorem, is given by a distribution of the form $p(x) \propto [1-(1-q) \beta x^2]^{1/(1-q)}$ with $\beta>0$. These distributions, sometimes referred to as $q$-Gaussians, are known to make, under appropriate constraints, extremal the functional $S_q$. Their $q=1$ and $q=2$ particular cases recover respectively Gaussian and Cauchy distributions. http://front.math.ucdavis.edu/cond-mat/0603593 --------------------------------------------------------------- 4162. A GEOMETRICAL STRUCTURE FOR AN INFINITE ORIENTED CLUSTER AND ITS UNIQUENESS Xian-Yuan Wu and Yu Zhang We consider the supercritical oriented percolation model. Let ${\fK}$ be all the percolation points. For each $u\in {\fK}$, we write $\gamma_u$ as its right-most path. Let $G=\cup_u \gamma_u$. In this paper, we show that $G$ is a single tree with only one topological end. We also present a relationship between ${\fK}$ and $G$ and construct a bijection between ${\fK}$ and $\Z$ using the preorder traversal algorithm. Through applications of this fundamental graph property, we show the uniqueness of an infinite oriented cluster by ignoring finite vertices. http://front.math.ucdavis.edu/math.PR/0603580 --------------------------------------------------------------- 4163. ULTRAMETRIC RANDOM FIELD A.Yu.Khrennikov and S.V.Kozyrev Gaussian random field on general ultrametric space is introduced as a solution of pseudodifferential stochastic equation. Covariation of the introduced random field is computed with the help of wavelet analysis on ultrametric spaces. Notion of ultrametric Markovianity, which describes independence of contributions to random field from different ultrametric balls is introduced. We show that the random field under investigation satisfies this property. http://front.math.ucdavis.edu/math.PR/0603584 --------------------------------------------------------------- 4164. ON RAW CODING OF CHAOTIC DYNAMICS Michael Blank We study raw coding of trajectories of a chaotic dynamical system by sequences of elements from a finite alphabet and show that there is a fundamental constraint on differences between codes corresponding to different trajectories of the dynamical system. http://front.math.ucdavis.edu/math.DS/0603575 --------------------------------------------------------------- 4165. CONDITIONED STABLE L\'{E}VY PROCESSES AND LAMPERTI REPRESENTATION Maria Emilia Caballero and Lo\"{i}c Chaumont (PMA) By killing a stable L\'{e}vy process when it leaves the positive half- line, or by conditioning it to stay positive, or by conditioning it to hit 0 continuously, we obtain three different positive self-similar Markov processes which illustrate the three classes described by Lamperti \cite{La}. For each of these processes, we compute explicitly the infinitesimal generator from which we deduce the characteristics of the underlying L\'{e}vy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable L\'{e}vy processes before their first passage time across level 0 which we describe here. As an application, we give the law of the minimum before an independent exponential time of a certain class of L \'{e}vy processes. It provides the explicit form of the spacial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of L\'{e}vy processes. http://front.math.ucdavis.edu/math.PR/0603613 --------------------------------------------------------------- 4166. SCATTERING LENGTH FOR STABLE PROCESSES B. Siudeja Let $\alpha\in(0,2)$ and $X_t$ be a symmetric $\alpha$-stable process. We define the scattering length $\Gamma(v)$ of the positive potential $v $ and prove several of its basic properties. We use the scattering length to findestimates for the first eigenvalue of the Schr\"odinger operator of the ``Neumann'' fractional Laplacian in a cube with potential $v$. http://front.math.ucdavis.edu/math.PR/0603627 --------------------------------------------------------------- 4167. ON THE NUMBER OF CIRCUITS IN RANDOM GRAPHS Enzo Marinari and Guilhem Semerjian We apply in this article (non rigorous) statistical mechanics methods to the problem of counting long circuits in graphs. The outcomes of this approach have two complementary flavours. On the algorithmic side, we propose an approximate counting procedure, valid in principle for a large class of graphs. On a more theoretical side, we study the typical number of long circuits in random graph ensembles, reproducing rigorously known results and stating new conjectures. http://front.math.ucdavis.edu/cond-mat/0603657 --------------------------------------------------------------- 4168. EXISTENCE OF SADDLE POINTS IN DISCRETE MARKOV GAMES AND ITS APPLICATION IN NUMERICAL METHODS FOR STOCHASTIC DIFFERENTIAL GAMES Q. S. Song and G. Yin This work establishes sufficient conditions for existence of saddle points in discrete Markov games. The result reveals the relation between dynamic games and static games using dynamic programming equations. This result enables us to prove existence of saddle points of non-separable stochastic differential games of regime-switching diffusions under appropriate conditions. http://front.math.ucdavis.edu/math.OC/0603600 --------------------------------------------------------------- 4169. BESSEL CONVOLUTIONS ON MATRIX CONES: ALGEBRAIC PROPERTIES AND RANDOM WALKS Michael Voit Bessel-type convolution algebras of bounded Borel measures on the matrix cones of positive semidefinite $q\times q$-matrices over $\mathbb R, \mathbb C, \mathbb H$ were introduced recently by R\"osler. These convolutions depend on some continuous parameter, generate commutative hypergroup structures and have Bessel functions of matrix argument as characters. Here, we first study the rich algebraic structure of these hypergroups. In particular, the subhypergroups and automorphisms are classified, and we show that each quotient by a subhypergroup carries a hypergroup structure of the same type. The algebraic properties are partially related to properties of random walks on matrix Bessel hypergroups. In particular, known properties of Wishart distributions, which form Gaussian convolution semigroups on these hypergroups, are put into a new light. Moreover, limit theorems for random walks on these hypergroups are presented. In particular, we obtain strong laws of large numbers and a central limit theorem with Wishart distributions as limits. http://front.math.ucdavis.edu/math.CA/0603017 --------------------------------------------------------------- 4170. LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY A FRACTIONAL BROWNIAN MOTION WITH HURST PARAMETER LESS THAN 1/2 Jorge A. Leon and Jaime San Martin In this paper we use the chaos decomposition approach to establish the existence of a unique continuous solution to linear fractional differential equations of the Skorohod type. Here the coefficients are deterministic, the inital condition is anticipating and the underlying fractional Brownian motion has Hurst parameter less than 1/2. We provide an explicit expression for the chaos decomposition of the solution in order to show our results. http://front.math.ucdavis.edu/math.PR/0603636 --------------------------------------------------------------- 4171. LIFETIME ASYMPTOTICS OF ITERATED BROWNIAN MOTION IN R^{N} Erkan Nane Let $\tau_{D}(Z) $ be the first exit time of iterated Brownian motion from a domain $D \subset \RR{R}^{n}$ started at $z\in D$ and let $P_{z}[\tau_ {D}(Z) >t]$ be its distribution. In this paper we establish the exact asymptotics of $P_{z}[\tau_{D}(Z) >t]$ over bounded domains as an improvement of the results in \cite{deblassie, nane2}, for $z\in D$ \begin{eqnarray} \lim_{t\to\infty} t^{-1/2}\exp({3/2}\pi^{2/3}\lambda_{D}^{2/3}t^ {1/3}) P_{z}[\tau_{D}(Z)>t]= C(z),\nonumber \end{eqnarray} where $C(z)=(\lambda_{D}2^{7/2})/\sqrt{3 \pi}(\psi(z)\int_{D}\psi(y)dy) ^{2} $. Here $\lambda_{D}$ is the first eigenvalue of the Dirichlet Laplacian ${1/2}\Delta$ in $D$, and $\psi $ is the eigenfunction corresponding to $\lambda_{D} $ . We also study lifetime asymptotics of Brownian-time Brownian motion (BTBM), $Z^{1}_{t}=z+X(|Y(t)|)$, where $X_{t}$ and $Y_{t}$ are independent one-dimensional Brownian motions. http://front.math.ucdavis.edu/math.PR/0603637 --------------------------------------------------------------- 4172. EDGEWORTH EXPANSION OF THE LARGEST EIGENVALUE DISTRIBUTION FUNCTION OF GUE AND LUE Leonard N. Choup We derive expansions of the Hermite and Laguerre kernels at the edge of the spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the finite n Laguerre Unitary Ensem- ble (LUEn), respectively. Using these large n kernel expansions, we prove an Edgeworth type theorem for the largest eigenvalue distribution function of GUEn and LUEn. In our Edgeworth expansion, the correction terms are expressed in terms of the same Painleve II function appearing in the leading term, i.e. in the Tracy-Widom distribution. We conclude with a brief discussion of the universality of these results. http://front.math.ucdavis.edu/math.PR/0603639 --------------------------------------------------------------- 4173. THE METASTABILITY THRESHOLD FOR MODIFIED BOOTSTRAP PERCOLATION IN D DIMENSIONS Alexander E. Holroyd In the modified bootstrap percolation model, sites in the cube {1,...,L}^d are initially declared active independently with probability p. At subsequent steps, an inactive site becomes active if it has at least one active nearest neighbour in each of the d dimensions, while an active site remains active forever. We study the probability that the entire cube is eventually active. For all d>=2 we prove that as L\to\infty and p\to 0 simultaneously, this probability converges to 1 if L=exp^{d-1} (lambda+epsilon)/p, and converges to 0 if L=exp^{d-1} (lambda-epsilon)/p, for any epsilon>0. Here exp^n denotes the n-th iterate of the exponential function, and the threshold lambda equals pi^2/6 for all d. http://front.math.ucdavis.edu/math.PR/0603645 --------------------------------------------------------------- 4174. LOG-CONCAVITY AND THE MAXIMUM ENTROPY PROPERTY OF THE POISSON DISTRIBUTION Oliver Johnson We prove that the Poisson distribution maximises entropy in the class of ultra-log-concave distributions, extending a result of Harremo\"{e}s. The proof uses ideas concerning log-concavity, and a semigroup action involving adding Poisson variables and thinning. We go on to show that the entropy is a concave function along this semigroup. http://front.math.ucdavis.edu/math.PR/0603647 --------------------------------------------------------------- 4175. QUENCHED NONEQUILIBRIUM CENTRAL LIMIT THEOREM FOR A TAGGED PARTICLE IN THE EXCLUSION PROCESS WITH BOND DISORDER M. D. Jara and C. Landim For a sequence of i.i.d. random variables $\{\xi_x : x\in \bb Z\}$ bounded above and below by strictly positive finite constants, consider the nearest-neighbor one-dimensional simple exclusion process in which a particle at $x$ (resp. $x+1$) jumps to $x+1$ (resp. $x$) at rate $\xi_x$. We examine a quenched nonequilibrium central limit theorem for the position of a tagged particle in the exclusion process with bond disorder $\{\xi_x : x\in \bb Z\}$. We prove that the position of the tagged particle converges under diffusive scaling to a Gaussian process if the other particles are initially distributed according to a Bernoulli product measure associated to a smooth profile $\rho_0:\bb R\to [0,1]$. http://front.math.ucdavis.edu/math.PR/0603653 --------------------------------------------------------------- 4176. ON DECOMPOSING RISK IN A FINANCIAL-INTERMEDIATE MARKET AND RESERVING Saul Jacka and Abdel Berkaoui We consider the problem of decomposing monetary risk in the presence of a fully traded market in {\it some} risks. We show that a mark-to- market approach to pricing leads to such a decomposition if the risk measure is time- consistent in the sense of Delbaen. http://front.math.ucdavis.edu/math.PR/0603041 --------------------------------------------------------------- 4177. ERGODIC THEORY FOR SDES WITH EXTRINSIC MEMORY M. Hairer and A. Ohashi We develop a theory of ergodicity for a class of random dynamical systems where the driving noise is not white. The two main tools of our analysis are the strong Feller property and topological irreducibility, introduced in this work for a class of non-Markovian systems. They allow us to obtain a criteria for ergodicity which is similar in nature to the Doob-Khas'minskii theorem. The second part of this article shows how it is possible to apply these results to the case of stochastic differential equations driven by fractional Brownian motion. It follows that under a non-degeneracy condition on the noise, such equations admit a unique adapted stationary solution. http://front.math.ucdavis.edu/math.PR/0603658 --------------------------------------------------------------- 4178. CORRECTION. CONNECT THE DOTS: HOW MANY RANDOM POINTS CAN A REGULAR CURVE PASS THROUGH? E. Arias-Castro and D. L. Donoho and X. Huo and C. A. Tovey Correction for Adv. in Appl. Probab. 37, no. 3 (2005), 571-603 http://front.math.ucdavis.edu/math.PR/0603673 --------------------------------------------------------------- 4179. LARGE DEVIATIONS FOR MANY BROWNIAN BRIDGES WITH SYMMETRISED INITIAL-TERMINAL CONDITION Stefan Adams and Wolfgang K\"onig Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$ with some non-degenerate initial measure on some fixed time interval $[0,\beta] $ with symmetrised initial-terminal condition. That is, for any $i$, the terminal location of the $i$-th motion is affixed to the initial point of the $\sigma(i)$-th motion, where $\sigma$ is a uniformly distributed random permutation of $1,...,N$. Such systems play an important role in quantum physics in the description of Boson systems at positive temperature $1/\beta$. In this paper, we describe the large-N behaviour of the empirical path measure (the mean of the Dirac measures in the $N$ paths) and of the mean of the normalised occupation measures of the $N$ motions in terms of large deviations principles. The rate functions are given as variational formulas involving certain entropies and Fenchel-Legendre transforms. Consequences are drawn for asymptotic independence statements and laws of large numbers. In the special case related to quantum physics, our rate function for the occupation measures turns out to be equal to the well-known Donsker- Varadhan rate function for the occupation measures of one motion in the limit of diverging time. This enables us to prove a simple formula for the large-N asymptotic of the symmetrised trace of ${\rm e}^{-\beta \mathcal{H}_N} $, where $\mathcal{H}_N$ is an $N$-particle Hamilton operator in a trap. http://front.math.ucdavis.edu/math.PR/0603702 --------------------------------------------------------------- 4180. FINITELY ADDITIVE BELIEFS AND UNIVERSAL TYPE SPACES Martin Meier The probabilistic type spaces in the sense of Harsanyi [Management Sci. 14 (1967/68) 159--182, 320--334, 486--502] are the prevalent models used to describe interactive uncertainty. In this paper we examine the existence of a universal type space when beliefs are described by finitely additive probability measures. We find that in the category of all type spaces that satisfy certain measurability conditions ($\kappa$-measurability, for some fixed regular cardinal $\kappa$), there is a universal type space (i.e., a terminal object) to which every type space can be mapped in a unique beliefs-preserving way. However, by a probabilistic adaption of the elegant sober-drunk example of Heifetz and Samet [Games Econom. Behav. 22 (1998) 260--273] we show that if all subsets of the spaces are required to be measurable, then there is no universal type space. http://front.math.ucdavis.edu/math.PR/0602656 --------------------------------------------------------------- 4181. THE TIME EVOLUTION OF PERMUTATIONS UNDER RANDOM STIRRING B\'alint Vet\H{o} We consider permutations of $\{1,...,n\}$ obtained by $\sqrt{nt}$ independent applications of random stirring. In each step the same marked stirring element is transposed with probability $1/n$ with any one of the $n$ elements. Normalizing by $\sqrt{n}$ we describe the asymptotic distribution of the cycle structure of these permutations, for all $t\ge0$, as $n\to\infty$. http://front.math.ucdavis.edu/math.PR/0603044 --------------------------------------------------------------- 4182. STATIONARITY OF PURE DELAY SYSTEMS AND QUEUES WITH IMPATIENT CUSTOMERS VIA STOCHASTIC RECURSIONS Pascal Moyal In this paper we solve a particular stochastic recursion in the stationary ergodic framework, and propose some applications of this result to the study of regenerativity (that is, finiteness of busy cycles) and stationarity of some queueing systems: pure delay systems, in which all customers are immediately served, and queues with impatient customers. In this latter case under the FIFO discipline, we prove as well the existence of a stationary workload on an enlarged probability space. http://front.math.ucdavis.edu/math.PR/0603709 --------------------------------------------------------------- 4183. ON THE ASYMPTOTIC DISTRIBUTION OF CERTAIN BIVARIATE REINSURANCE TREATIES Enkelejd Hashorva Let (X_n,Y_n), n\ge 1 be bivariate random claim sizes with common distribution function F and let N(t), t \ge 0 be a stochastic process which counts the number of claims that occur in the time interval [0,t], t \ge 0. In this paper we derive the joint asymptotic distribution of randomly indexed order statistics of the random sample (X_1,Y_1),(X_2,Y_2),...,(X_{N(t)},Y_{N(t)}) which is then used to obtain asymptotic representations for the joint distribution of two generalised largest claims reinsurance treaties available under specific insurance settings. As a by-product we obtain a stochastic representation of a m-dimensional Lambda-extremal variate in terms of iid unit exponential random variables. http://front.math.ucdavis.edu/math.PR/0603719 --------------------------------------------------------------- 4184. THE ZEROS OF GAUSSIAN RANDOM HOLOMORPHIC FUNCTIONS ON $\C^N$, AND HOLE PROBABILITY Scott Zrebiec We consider a class of Gaussian random holomorphic functions, whose expected zero set is uniformly distributed over $\C^n $. This class is unique (up to multiplication by a non zero holomorphic function), and is closely related to a Gaussian field over a Hilbert space of holomorphic functions on the reduced Heisenberg group. For a fixed random function of this class, we show that the probability that there are no zeros in a ball of large radius, is less than $e^{-c_1 r^{2n+2}}$, and is also greater than $e^{-c_2 r^{2n+2}}$. Enroute to this result we also compute probability estimates for the event that a random function's unintegrated counting function deviates significantly from its mean. http://front.math.ucdavis.edu/math.CV/0603696 --------------------------------------------------------------- 4185. EXCHANGEABLE PARTITIONS DERIVED FROM MARKOVIAN COALESCENTS Rui Dong and Alexander Gnedin and Jim Pitman Kingman derived the Ewens sampling formula for random partitions describing the genetic variation in a neutral mutation model defined by a Poisson process of mutations along lines of descent governed by a simple coalescent process, and observed that similar methods could be applied to more complex models. M{\"o}hle described the recursion which determines the generalization of the Ewens sampling formula in the situation when the lines of descent are governed by a $\Lambda$-coalescent, which allows multiple mergers. Here we show that the basic integral representation of transition rates for the $\Lambda$- coalescent is forced by sampling consistency under more general assumptions on the coalescent process. Exploiting an analogy with the theory of regenerative partition structures, we provide various characterizations of the associated partition structures in terms of discrete-time Markov chains. http://front.math.ucdavis.edu/math.PR/0603745 --------------------------------------------------------------- 4186. BEHAVIOR OF THE EULER SCHEME WITH DECREASING STEP IN A DEGENERATE SITUATION Vincent Lemaire (LAMA) The aim of this paper is to study the behavior of the weighted empirical measures of the decreasing step Euler scheme of a one-dimensional diffusion process having multiple invariant measures. This situation can occur when the drift and the diffusion coefficient are vanish simultaneously. As a first step, we give a brief description of the Feller's classification of the one-dimensional process. We recall the concept of attractive and repulsive boundary point and introduce the concept of strongly repulsive point. That allows us to establish a classification of the ergodic behavior of the diffusion. We conclude this section by giving necessary and sufficient conditions on the nature of boundary points in terms of Lyapunov functions. In the second section we use this characterization to study the decreasing step Euler scheme. We give also an numerical example in higher dimension. http://front.math.ucdavis.edu/math.PR/0604021 --------------------------------------------------------------- 4187. INVASION AND ADAPTIVE EVOLUTION FOR INDIVIDUAL-BASED SPATIALLY STRUCTURED POPULATIONS Nicolas Champagnat (WIAS) and Sylvie M\'{e}l\'{e}ard (MODAL'X and FESE) The interplay between space and evolution is an important issue in population dynamics, that is in particular crucial in the emergence of polymorphism and spatial patterns. Recently, biological studies suggest that invasion and evolution are closely related. Here we model the interplay between space and evolution starting with an individual-based approach and show the important role of parameter scalings on clustering and invasion. We consider a stochastic discrete model with birth, death, competition, mutation and spatial diffusion, where all the parameters may depend both on the position and on the trait of individuals. The spatial motion is driven by a reflected diffusion in a bounded domain. The interaction is modelled as a trait competition between individuals within a given spatial interaction range. First, we give an algorithmic construction of the process. Next, we obtain large population approximations, as weak solutions of nonlinear reaction-diffusion equations with Neumann's boundary conditions. As the spatial interaction range is fixed, the nonlinearity is nonlocal. Then, we make the interaction range decrease to zero and prove the convergence to spatially localized nonlinear reaction- diffusion equations, with Neumann's boundary conditions. Finally, simulations based on the microscopic individual-based model are given, illustrating the strong effects of the spatial interaction range on the emergence of spatial and phenotypic diversity (clustering and polymorphism) and on the interplay between invasion and evolution. The simulations focus on the qualitative differences between local and nonlocal interactions. http://front.math.ucdavis.edu/math.PR/0604041 --------------------------------------------------------------- 4188. PROCESSES WITH INERT DRIFT David White We construct a stochastic process whose drift is a function of the process's local time at a reflecting barrier. The process arose as a model of the interactions of a Brownian particle and an inert particle in \citep {knight:01}. Interesting asymptotic results are obtained for two different arrangements of inert particles and Brownian particles. A version of the process in $ \Re^d$ is also constructed. http://front.math.ucdavis.edu/math.PR/0604052 --------------------------------------------------------------- 4189. WHEN THE LAW OF LARGE NUMBERS FAILS FOR INCREASING SUBSEQUENCES OF RANDOM PERMUTATIONS Ross G. Pinsky Let the random variable $Z_{n,k}$ denote the number of increasing subsequences of length $k$ in a random permutation from $S_n$, the symmetric group of permutations of $\{1,...,n\}$. In a recent paper (http://front.math.ucdavis.edu/math.PR/0407353) we showed that the weak law of large numbers holds for $Z_{n,k_n}$ if $k_n=o(n^\frac25)$; that is, $$ \lim_{n\to\infty}\frac{Z_{n,k_n}} {EZ_{n,k_n}}=1, \text{in probability}. $$ The method of proof employed there used the second moment method and demonstrated that this method cannot work if the condition $k_n=o(n^\frac25)$ does not hold. It follows from results concerning the longest increasing subsequence of a random permutation that the law of large numbers cannot hold for $Z_ {n,k_n}$ if $k_n\ge cn^\frac12$, with $c>2$. Presumably there is a critical exponent $l_0$ such that the law of large numbers holds if $k_n=O(n^l)$, with $l0$, for some $l>l_0$. Several phase transitions concerning increasing subsequences occur at $l=\frac12$, and these would suggest that $l_0=\frac12$. However, in this paper, we show that the law of large numbers fails for $Z_{n,k_n}$ if $\limsup_{n\to\infty}\frac{k_n}{n^\frac49}=\infty$. Thus the critical exponent, if it exists, must satisfy $l_0\in[\frac25,\frac49]$. http://front.math.ucdavis.edu/math.PR/0604067 --------------------------------------------------------------- 4190. A SIMPLE FLUCTUATION LOWER BOUND FOR A DISORDERED MASSLESS RANDOM CONTINUOUS SPIN MODEL IN D=2 C. Kuelske and E. Orlandi We prove a finite volume lower bound of the order of the squareroot of log N on the delocalization of a disordered continuous spin model (resp. effective interface model) in d = 2 in a box of size N . The interaction is assumed to be massless, possibly anharmonic and dominated from above by a Gaussian. Disorder is entering via a linear source term. For this model delocalization with the same rate is proved to take place already without disorder, so our proof shows that randomness will only enhance fluctuations. http://front.math.ucdavis.edu/math.PR/0604068 --------------------------------------------------------------- 4191. THE MAXIMUM OF THE LOCAL TIME OF A DIFFUSION PROCESS IN A DRIFTED BROWNIAN POTENTIAL Alexis Devulder (PMA) We consider a one-dimensional diffusion process in a drifted Brownian potential. We are interested in the maximum of its local time, and study its almost sure asymptotic behaviour, which is proved to be different from the behaviour of the maximum local time of the transient random walk in random environment. http://front.math.ucdavis.edu/math.PR/0604078 --------------------------------------------------------------- 4192. CAVITY METHOD IN THE SPHERICAL SK MODEL Dmitry Panchenko We develop the cavity method for the spherical Sherrington- Kirkpatrick model at high temperature and small external field. As one application, we carry out the second moment computations for the overlap and the magnetization. http://front.math.ucdavis.edu/math.PR/0604081 --------------------------------------------------------------- 4193. ON THE OVERLAP IN THE MULTIPLE SPHERICAL SK MODELS Dmitry Panchenko and Michel Talagrand In order to study certain questions concerning the distribution of the overlap in Sherrington-Kirkpatrick type models, such as the chaos and ultrametricity problems, it seems natural to study the free energy of multiple systems with constrained overlaps. One can write analogues of Guerra's replica symmetry breaking bound for such systems but it is not at all obvious how to choose informative functional order parameters in these bounds. We were able to make some progress for spherical pure $p$-spin SK models where many computations can be made explicitly. For pure 2-spin model we prove ultrametricity and chaos in an external field. For the pure $p$-spin model for even $p>4$ without an external field we describe two possible values of the overlap of two systems at different temperatures. We also prove a somewhat unexpected result which shows that in the 2-spin model the support of the joint overlap distribution is not always witnessed at the level of the free energy and,for example, ultrametricity holds only in a weak sense. http://front.math.ucdavis.edu/math.PR/0604082 --------------------------------------------------------------- 4194. DERIVATIVES OF ENTROPY RATE IN SPECIAL FAMILIES OF HIDDEN MARKOV CHAINS Guangyue Han and Brian Marcus Consider a hidden Markov chain obtained as the observation process of an ordinary Markov chain corrupted by noise. Zuk, et. al. [13], [14] showed how, in principle, one can explicitly compute the derivatives of the entropy rate of at extreme values of the noise. Namely, they showed that the derivatives of standard upper approximations to the entropy rate actually stabilize at an explicit finite time. We generalize this result to a natural class of hidden Markov chains called ``Black Holes.'' We also discuss in depth special cases of binary Markov chains observed in binary symmetric noise, and give an abstract formula for the first derivative in terms of a measure on the simplex due to Blackwell. http://front.math.ucdavis.edu/cs.IT/0603059 --------------------------------------------------------------- 4195. THE CODING OF COMPACT REAL TREES BY REAL VALUED FUNCTIONS Thomas Duquesne This paper is a detailled study of the coding of real trees by real valued functions that is motivated by probabilistic problems related to continuum random trees. Indeed it is known since the works of Aldous (1993) and Le Gall (1991) that a continuous non-negative function $h$ on $[0,1]$ such that $h(0)=0$ can be seen as the contour process of a compact real tree. This particular coding of a compact real tree provides additional structures, namely a root that is the vertex corresponding to $0\in [0,1]$, a linear order inherited from the usual order on $[0,1]$ and a measure induced by the Lebesgue measure on $[0,1]$; of course, the root, the linear order and the measure obtained by such a coding have to satisfy some compatibility conditions. In this paper, we prove that any compact real tree equipped with a root, a linear order and a measure that are compatible can be encoded by a non-negative function $h$ defined on a finite interval $[0, M]$, that is assumed to be left-continuous with right-limit, without positive jump and such that $h(0+)=h(0)=0$. Moreover, this function is unique if we assume that the exploration of the tree induced by such a coding backtracks as less as possible. We also prove that a measure-change on the tree corresponds to a re-parametrization of the coding function. In addition, we describe several path-properties of the coding function in terms of the metric properties of the real tree. http://front.math.ucdavis.edu/math.PR/0604106 --------------------------------------------------------------- 4196. ON THE FUTURE INFIMUM OF POSITIVE SELF-SIMILAR MARKOV PROCESSES J.C. Pardo We establish integral tests and laws of the iterated logarithm for the upper envelope of the future infimum of positive self-similar Markov processes and for increasing self-similar Markov processes at 0 and infinity. Our proofs are based on the Lamperti representation and time reversal arguments due to Chaumont and Pardo [9]. These results extend laws of the iterated logarithm for the future infimum of Bessel processes due to Khoshnevisan et al. [11]. http://front.math.ucdavis.edu/math.PR/0604110 --------------------------------------------------------------- 4197. LAWS AND LIKELIHOODS FOR ORNSTEIN UHLENBECK-GAMMA AND OTHER BNS OU STOCHASTIC VOLATILTY MODELS WITH EXTENSIONS Lancelot F. James In recent years there have been many proposals as flexible alternatives to Gaussian based continuous time stochastic volatility models. A great deal of these models employ positive L\'evy processes. Among these are the attractive non-Gaussian positive Ornstein-Uhlenbeck (OU) processes proposed by Barndorff-Nielsen and Shephard (BNS) in a series of papers. One current problem of these approaches is the unavailability of a tractable likelihood based statistical analysis for the returns of financial assets. This paper, while focusing on the BNS models, develops general theory for the implementation of statistical inference for a host of models. Specifically we show how to reduce the infinite-dimensional process based models to finite, albeit high, dimensional ones. Inference can then be based on Monte Carlo methods. As highlights, specific to BNS we show that an OU process driven by an infinite activity Gamma process, that is an OU-$\Gamma$, exhibits unique features which allows one to exactly sample from relevant joint distributions. We show that this is a consequence of the OU structure and the unique calculus of Gamma and Dirichlet processes. Owing to another connection between Gamma/Dirichlet processes and the theory of Generalized Gamma Convolutions (GGC) we identify a large class of models, we call (FGGC), where one can perfectly sample marginal distributions relevant to option pricing and Monte Carlo likelihood analysis. This involves a curious result, we establish as Theorem 6.1. We also discuss analytic techniques and candidate densities for Monte-Carlo procedures which can be applied to more general http://front.math.ucdavis.edu/math.ST/0604086 --------------------------------------------------------------- 4198. MAXIMUM PRINCIPLE FOR SPDES AND ITS APPLICATIONS N.V. Krylov The maximum principle for SPDEs is established in multidimensional $C^ {1}$ domains. An application is given to proving the H\"older continuity up to the boundary of solutions of one-dimensional SPDEs. http://front.math.ucdavis.edu/math.PR/0604125 --------------------------------------------------------------- 4199. A FAMILY OF NON-GAUSSIAN MARTINGALES WITH GAUSSIAN MARGINALS kais Hamza and Fima C. Klebaner We construct a family of non-Gaussian martingales the marginals of which are all Gaussian. We give the predictable quadratic variation of these processes and show they do not have continuous paths. These processes are Markovian and inhomogeneous in time, and we give their infinitesimal generators. Within this family we find a class of piecewise deterministic pure jump processes and describe the laws of jumps and times between the jumps. http://front.math.ucdavis.edu/math.PR/0604127 --------------------------------------------------------------- 4200. STOCHASTIC EQUATIONS WITH TIME-DEPENDENT DRIFT DRIVEN BY LEVY PROCESSES V.P.Kurenok Using the method of Krylov's estimates, we prove the existence of weak solutions of stochastic differential equations driven by purely discontinuous Levy processes satisfying an additional assumption. The diffusion coefficient is assumed to be one and the time-dependent drift is measurable and bounded. http://front.math.ucdavis.edu/math.PR/0604136 --------------------------------------------------------------- 4201. CONDITIONED GALTON-WATSON TREES DO NOT GROW Svante Janson An example is given which shows that, in general, conditioned Galton- Watson trees cannot be obtained by adding vertices one by one, as has been shown in a special case by Luczak and Winkler. http://front.math.ucdavis.edu/math.PR/0604141 --------------------------------------------------------------- 4202. SEMI-SELFDECOMPOSABLE LAWS IN THE MINIMUM SCHEME S Satheesh and E Sandhya We discuss semi-selfdecomposable laws in the minimum scheme and characterize them using an autoregressive model. Semi-Pareto and semi-Weibull laws of Pillai (1991) are shown to be semi-selfdecomposable in this scheme. Methods for deriving this class of laws are then attempted from the angle of randomization. Finally, discrete analogues of these results are also considered. http://front.math.ucdavis.edu/math.PR/0604146 --------------------------------------------------------------- 4203. UNIFORM FORMULAE FOR COEFFICIENTS OF MEROMORPHIC FUNCTIONS IN TWO VARIABLES. PART I Manuel Lladser Uniform asymptotic formulae for arrays of complex numbers of the form $(f_{r,s})$, with $r$ and $s$ nonnegative integers, are provided as $r $ and $s$ converge to infinity at a comparable rate. Our analysis is restricted to the case in which the generating function $F(z,w):=\sum f_{r,s} z^r w^s$ is meromorphic in a neighborhood of the origin. We provide uniform asymptotic formulae for the coefficients $f_{r,s}$ along directions in the $(r,s) $-lattice determined by regular points of the singular variety of $F$. Our main result derives from the analysis of a one dimensional parameter-varying integral describing the asymptotic behavior of $f_{r,s}$. We specifically consider the case in which the phase term of this integral has a unique stationary point, however, allowing the possibility that one or more stationary points of the amplitude term coalesce with this. Our results find direct application in certain problems associated to the Lagrange inversion formula as well as bivariate generating functions of the form $v(z)/(1-w\cdot u(z))$. http://front.math.ucdavis.edu/math.CO/0604152 --------------------------------------------------------------- 4204. DESIGN FLAWS IN THE IMPLEMENTATION OF THE ZIGGURAT AND MONTY PYTHON METHODS (AND SOME REMARKS ON MATLAB RANDN) Boaz Nadler {\em Ziggurat} and {\em Monty Python} are two fast and elegant methods proposed by Marsaglia and Tsang to transform uniform random variables to random variables with normal, exponential and other common probability distributions. While the proposed methods are theoretically correct, we show that there are various design flaws in the uniform pseudo random number generators (PRNG's) of their published implementations for both the normal and Gamma distributions \cite{Ziggurat,{Gamma},Monty}. These flaws lead to non-uniformity of the resulting pseudo-random numbers and consequently to noticeable deviations of their outputs from the required distributions. In addition, we show that the underlying uniform PRNG of the published implementation of Matlab's \texttt{randn}, which is also based on the Ziggurat method, is not uniformly distributed with correlations between consecutive pairs. Also, we show that the simple linear initialization of the registers in matlab's \texttt {randn} may lead to non-trivial correlations between output sequences initialized with different (related or even random unrelated) seeds. These, in turn, may lead to erroneous results for stochastic simulations. http://front.math.ucdavis.edu/math.ST/0603058 --------------------------------------------------------------- 4205. EFFECTIVE BANDWIDTH PROBLEM REVISITED Vyacheslav M. Abramov The paper studies a single-server queueing system with autonomous service and $\ell$ priority classes. Arrival and departure processes are defined by marked point processes. There are $\ell$ buffers corresponding to priority classes, and upon arrival a unit of the $k$th priority class occupies the place in the $k$th buffer. Let $N^{(k)}$, $k=1,2,...,\ell$ denote the quota for the total $k$th buffer content. The values $N^{(k)}$ are assumed to be large, and queueing systems both with finite and infinite buffers are studied. In the case of system with finite buffers, the values $N^{(k)}$ characterize buffer capacities. The paper discusses a circle of problems related to optimization of performance measures associated with overflowing the quota of buffer contents. Our approach to this problem is new, and presentation of our results is simple and clear for real applications. http://front.math.ucdavis.edu/math.PR/0604182 --------------------------------------------------------------- 4206. LIMITING BEHAVIOR OF THE DISTANCE OF A RANDOM WALK Nathanael Berestycki and Rick Durrett This investigation is motivated by a result we proved recently for the random transposition random walk: the distance from the starting point of the walk has a phase transition from a linear regime to a sublinear regime at time $n/2$. Here, we study three new examples. It is trivial that the distance for random walk on the hypercube is smooth and is given by one simple formula. In the case of random adjacent transpositions, we find that there is no phase transition even though the distance has different scalings in three different regimes. In the case of a random 3-regular graph, there is a phase transition from linear growth to a constant equal to the diameter of the graph, at time $3 \log_2 n$. http://front.math.ucdavis.edu/math.PR/0604188 --------------------------------------------------------------- 4207. HEAVY TAILS IN LAST-PASSAGE PERCOLATION Ben Hambly and James B. Martin We consider last-passage percolation models in two dimensions, in which the underlying weight distribution has a heavy tail of index alpha<2. We prove scaling laws and asymptotic distributions, both for the passage times and for the shape of optimal paths; these are expressed in terms of a family (indexed by alpha) of "continuous last-passage percolation" models in the unit square. In the extreme case alpha=0 (corresponding to a distribution with slowly varying tail) the asymptotic distribution of the optimal path can be represented by a random self-similar measure on [0,1], whose multifractal spectrum we compute. By extending the continuous last-passage percolation model to R^2 we obtain a heavy-tailed analogue of the Airy process, representing the limit of appropriately scaled vectors of passage times to different points in the plane. We give corresponding results for a directed percolation problem based on alpha-stable Levy processes, and indicate extensions of the results to higher dimensions. http://front.math.ucdavis.edu/math.PR/0604189 --------------------------------------------------------------- 4208. TWO NON-REGULAR EXTENSIONS OF LARGE DEVIATION BOUND Masahito Hayashi We formulate two types of extensions of the large deviation theory initiated by Bahadur in a non-regular setting. One can be regarded as a bound of the point estimation, the other can be regarded as the limit of a bound of the interval estimation. Both coincide in the regular case, but do not necessarily coincide in a non-regular case. Using the limits of relative R\'{e}nyi entropies, we derive their upper bounds and give a necessary and sufficient condition for the coincidence of the two upper bounds. We also discuss the attainability of these two bounds in several non-regular location shift families. http://front.math.ucdavis.edu/math.PR/0604197 --------------------------------------------------------------- 4209. ATTRACTING EDGE AND STRONGLY EDGE REINFORCED WALKS V. Limic and P. Tarres The goal is to show that an edge reinforced random walk on a graph of bounded degree, with reinforcement {\em weight function} $W$ taken from a general class of reciprocally summable reinforcement weight functions, traverses a random {\em attracting} edge at all large times. The statement of the main theorem is very close to settling the original conjecture of Sellke (1994). An important corollary of this main result says that if $W$ is reciprocally summable and nondecreasing, the attracting edge exists on any graph of bounded degree, with probability 1. Another corollary is the main theorem of Limic (2003) where the class of weights was restricted to reciprocally summable powers. The proof uses martingale and other techniques developed by the authors in separate studies of edge and vertex reinforced walks (Limic (2003), Tarr\`es (2004)), and of nonconvergence properties of stochastic algorithms towards unstable equilibrium points of the associated deterministic dynamics, Tarr\`es (2000). http://front.math.ucdavis.edu/math.PR/0604200 --------------------------------------------------------------- 4210. THE MOMENT PROBLEM AND THE WIENER SPACE Frederik S Herzberg Consider an $L^1$-continuous functional $\ell$ on the vector space of polynomials of Brownian motion at given times, suppose $\ell $ commutes with the quadratic variation in a natural sense, and consider a finite set of polynomials of Brownian motion at rational times, $p_1(\vec b),...,p_m,(\vec b)$, mapping the Wiener space to $\mathbb{R}$. Similarly to the moment problem for a finite-dimensional space of polynomials, we give sufficient conditions under which $\ell$ can be written in the form $\int \cdot d\mu$ for some finite measure $\mu$ on the Wiener space such that $\mu$-almost surely, all the random variables $p_1(\vec b),...,p_m,(\vec b)$ are nonnegative. http://front.math.ucdavis.edu/math.PR/0604211 --------------------------------------------------------------- 4211. PROCESSOR SHARING QUEUES WITH IMPATIENCE Christian H. Gromoll (STANFORD-MATHS) and Philippe Robert (INRIA Rocquencourt), Bert Zwart (TUE) We investigate a processor sharing queue with renewal arrivals and generally distributed service times. Impatient jobs may abandon the queue, or renege, before completing service. The corresponding stochastic processes are represented by measure valued Markov processes on R^2\_+. A scaling procedure that gives rise to a fluid model with a nontrivial, yet tractable steady state behavior, is presented. This fluid model model captures many essential features of the underlying stochastic model, and it is used to analyze the impact of impatience in processor sharing queues. http://front.math.ucdavis.edu/math.PR/0604215 --------------------------------------------------------------- 4212. HEURISTICS FOR THE WHITEHEAD MINIMIZATION PROBLEM R.M. Haralick and A.D. Miasnikov and A.G. Myasnikov In this paper we discuss several heuristic strategies which allow one to solve the Whitehead's minimization problem much faster (on most inputs) than the classical Whitehead algorithm. The mere fact that these strategies work in practice leads to several interesting mathematical conjectures. In particular, we conjecture that the length of most non-minimal elements in a free group can be reduced by a Nielsen automorphism which can be identified by inspecting the structure of the corresponding Whitehead Graph. http://front.math.ucdavis.edu/math.GR/0604204 --------------------------------------------------------------- 4213. RIGOROUS INEQUALITIES BETWEEN LENGTH AND TIME SCALES IN GLASSY SYSTEMS Andrea Montanari and Guilhem Semerjian Glassy systems are characterized by an extremely sluggish dynamics without any simple sign of long range order. It is a debated question whether a correct description of such phenomenon requires the emergence of a large correlation length. We prove rigorous bounds between length and time scales implying the growth of a properly defined length when the relaxation time increases. Our results are valid in a rather general setting, which covers finite- dimensional and mean field systems. As an illustration, we discuss the Glauber (heat bath) dynamics of p-spin glass models on random regular graphs. We present the first proof that a model of this type undergoes a purely dynamical phase transition not accompanied by any thermodynamic singularity. http://front.math.ucdavis.edu/cond-mat/0603018 --------------------------------------------------------------- 4214. A HYBRID SEARCH ALGORITHM FOR THE WHITEHEAD MINIMIZATION PROBLEM A.D. Myasnikov and R.M Haralick The Whitehead Minimization problem is a problem of finding elements of the minimal length in the automorphic orbit of a given element of a free group. The classical algorithm of Whitehead that solves the problem depends exponentially on the group rank. Moreover, it can be easily shown that exponential blowout occurs when a word of minimal length has been reached and, therefore, is inevitable except for some trivial cases. In this paper we introduce a deterministic Hybrid search algorithm and its stochastic variation for solving the Whitehead minimization problem. Both algorithms use search heuristics that allow one to find a length- reducing automorphism in polynomial time on most inputs and significantly improve the reduction procedure. The stochastic version of the algorithm employs a probabilistic system that decides in polynomial time whether or not a word is minimal. The stochastic algorithm is very robust. It has never happened that a non-minimal element has been claimed to be minimal. http://front.math.ucdavis.edu/math.GR/0604206 --------------------------------------------------------------- 4215. ANALYSE NON STANDARD DU BRUIT Michel Fliess (LIX and INRIA Futurs) Thanks to the nonstandard formalization of fast oscillating functions, due to P. Cartier and Y. Perrin, an appropriate mathematical framework is derived for new non-asymptotic estimation techniques, which do not necessitate any statistical analysis of the noises corrupting any sensor. Various applications are deduced for multiplicative noises, for the length of the parametric estimation windows, and for burst errors. http://front.math.ucdavis.edu/cs.CE/0603003 --------------------------------------------------------------- 4216. RECURRENCE OF RANDOM WALK TRACES Itai Benjamini and Ori Gurel-Gurevich and Russell Lyons We show that the edges crossed by a random walk in a network form a recurrent graph a.s. In fact, the same is true when those edges are weighted by the number of crossings. http://front.math.ucdavis.edu/math.PR/0603060 --------------------------------------------------------------- 4217. OPERATOR SCALING STABLE RANDOM FIELDS Hermine Bierm\'{e} (MAP5) and Mark M. Meerschaert and Hans-Peter Scheffler A scalar valued random field is called operator-scaling if it satisfies a self-similarity property for some matrix E with positive real parts of the eigenvalues. We present a moving average and a harmonizable representation of stable operator scaling random fields by utilizing so called E- homogeneous functions. These fields also have stationary increments and are stochastically continuous. In the Gaussian case critical H\"{o}lder-exponents and the Hausdorff-dimension of the sample paths are also obtained. http://front.math.ucdavis.edu/math.PR/0602664 --------------------------------------------------------------- 4218. PROCESSES ON UNIMODULAR RANDOM NETWORKS David Aldous and Russell Lyons We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasi-transitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasi-transitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications to stochastic comparison of continuous-time random walk. http://front.math.ucdavis.edu/math.PR/0603062 --------------------------------------------------------------- 4219. MULTICRITICAL CONTINUOUS RANDOM TREES J. Bouttier and P. Di Francesco and E. Guitter We introduce generalizations of Aldous' Brownian Continuous Random Tree as scaling limits for multicritical models of discrete trees. These discrete models involve trees with fine-tuned vertex-dependent weights ensuring a k-th root singularity in their generating function. The scaling limit involves continuous trees with branching points of order up to k+1. We derive explicit integral representations for the average profile of this k-th order multicritical continuous random tree, as well as for its history distributions measuring multi-point correlations. The latter distributions involve non-positive universal weights at the branching points together with fractional derivative couplings. We prove universality by rederiving the same results within a purely continuous axiomatic approach based on the resolution of a set of consistency relations for the multi-point correlations. The average profile is shown to obey a fractional differential equation whose solution involves hypergeometric functions and matches the integral formula of the discrete approach. http://front.math.ucdavis.edu/math-ph/0603007 --------------------------------------------------------------- 4220. RANDOM ENERGY MODEL WITH COMPACT DISTRIBUTIONS Nabin Kumar Jana In this paper we study the Random energy model - so called toy model of the spin glass theory - where the underlying distributions are compactly supported. We prove a general theorem on the asymptotics of free energy and obtain formulae in several interesting cases - like uniform distribution, truncated double exponential. http://front.math.ucdavis.edu/math.PR/0602666 --------------------------------------------------------------- 4221. THRESHOLD $THETA GEQ 2$ CONTACT PROCESSES ON HOMOGENEOUS TREES Luiz Renato Fontes and Roberto H. Schonmann We study the threshold $theta geq 2$ contact process on a homogeneous tree $T_b$ of degree $kappa = b + 1$, with infection parameter $lambda geq 0$ and started from a product measure with density $p$. The corresponding mean-field model displays a discontinuous transition at a critical point $lambda_c^{MF}(kappa,theta)$ and for $lambda geq lambda_c^{MF} (kappa,theta)$ it survives iff $p geq p_c^{MF}(kappa,theta,lambda)$, where this critical density satisfies $0 < p_c^{MF}(kappa,theta,lambda) < 1$, $lim_{lambda to infty} p_c^{MF}(kappa,theta,lambda) = 0$. For large $b$, we show that the process on $T_b$ has a qualitatively similar behavior when $lambda$ is small, including the behavior at and close to the critical point $lambda_c(T_b,theta) $. In contrast, for large $lambda$ the behavior of the process on $T_b$ is qualitatively distinct from that of the mean-field model in that the critical density has $p_c(T_b,theta,infty) := lim_{lambda to infty} p_c(T_b,theta,lambda) > 0$. We also show that $lim_{b to infty} b lambda_c(T_b,theta) = Phi_{theta}$, where $1 < Phi_2 < Phi_3 < ...$, $lim_{theta to infty} Phi_{theta} = infty$, and $0 < liminf_{b to infty} b^{theta(theta-1)} p_c(T_b,theta,infty) leq limsup_{b to infty} b^{theta/(theta-1)} p_c(T_b,theta,infty) < infty$. http://front.math.ucdavis.edu/math.PR/0603109 --------------------------------------------------------------- 4222. STOCHASTIC EQUATION ON COMPACT GROUPS IN DISCRETE NEGATIVE TIME Jir\^o Akahori and Chihiro Uenishi and Kouji Yano In this paper a stochastic equation on compact groups in discrete negative time is studied. This is closely related to Tsirelson's stochastic differential equation, of which any solution is non-strong. How the group action reflects on the set of solutions is investigated. It is applied to generalize Yor's result and give a necessary and sufficient condition for existence of a strong solution and for uniqueness in law. http://front.math.ucdavis.edu/math.PR/0603113 --------------------------------------------------------------- 4223. TRANSLATION-INVARIANCE OF TWO-DIMENSIONAL GIBBSIAN SYSTEMS OF PARTICLES WITH INTERNAL DEGREES OF FREEDOM Thomas Richthammer The conservation of translation as a symmetry in two-dimensional systems with interaction is a classical subject of statistical mechanics. Here we establish such a result for Gibbsian systems of marked particles with two-body interaction, where the interesting cases of singular, hard-core and discontinuous interaction are included. http://front.math.ucdavis.edu/math.PR/0603140 --------------------------------------------------------------- 4224. REGULAR VARIATION AND SMILE ASYMPTOTICS Shalom Benaim and Peter Friz We consider risk-neutral returns and show how their tail asymptotics translate directly to asymptotics of the implied volatility smile, thereby sharpening Roger Lee's celebrated moment formula. The theory of regular variation provides the ideal mathematical framework to formulate and prove such results. The practical value of our formulae comes from the vast literature on tail asymptotics and our conditions are often seen to be true by simple inspection of known results. http://front.math.ucdavis.edu/math.PR/0603146 --------------------------------------------------------------- 4225. SOME PARABOLIC PDES WHOSE DRIFT IS AN IRREGULAR RANDOM NOISE IN SPACE Francesco Russo (LAGA) and Gerald Trutnau (SFB 343) We consider a new class of random partial differential equation of parabolic type where the stochastic term is constituted by an irregular noisy drift, not necessarily Gaussian. We provide a suitable interpretation and we study existence. After freezing a realization of the drift (stochastic process), we study existence and uniqueness (in some suitable sense) of the associated parabolic equation and we investigate probabilistic interpretation. http://front.math.ucdavis.edu/math.PR/0602669 --------------------------------------------------------------- 4226. THE LEBESGUE DECOMPOSITION OF THE FREE ADDITIVE CONVOLUTION OF TWO PROBABILITY DISTRIBUTIONS Serban Teodor Belinschi We prove that the free additive convolution of two Borel probability measures supported on the real line can have a component that is singular continuous with respect to the Lebesgue measure on the real line only if one of the two measures is a point mass. The density of the absolutely continuous part with respect to the Lebesgue measure is shown to be analytic wherever positive and finite. The atoms of the free additive convolution of Borel probability measures on the real line have been described by Bercovici and Voiculescu in a previous paper. http://front.math.ucdavis.edu/math.OA/0603104 --------------------------------------------------------------- 4227. EXPONENTIAL RANDOM ENERGY MODEL Nabin Kumar Jana In this paper the Random Energy Model(REM) under exponential type environment is considered which includes double exponential and Gaussian cases. Limiting Free Energy is evaluated in these models. Limiting Gibbs' distribution is evaluated in the double exponential case. http://front.math.ucdavis.edu/math.PR/0602670 --------------------------------------------------------------- 4228. 2-FOLD AND 3-FOLD MIXING: WHY 3-DOT-TYPE COUNTEREXAMPLES ARE IMPOSSIBLE IN ONE DIMENSION Thierry De La Rue (LMRS) V.A. Rohlin asked in 1949 whether 2-fold mixing implies 3-fold mixing for a stationary process indexed by Z, and the question remains open today. In 1978, F. Ledrappier exhibited a counterexample to the 2-fold mixing implies 3-fold mixing problem, the so-called "3-dot system", but in the context of stationary random fields indexed by ZxZ. In this work, we first present an attempt to adapt Ledrappier's construction to the one-dimensional case, which finally leads to a stationary process which is 2-fold but not 3-fold mixing conditionally to the sigma-algebra generated by some factor process. Then, using arguments coming from the theory of joinings, we will give some strong obstacles proving that Ledrappier's counterexample can not be fully adapted to one-dimensional stationary processes. http://front.math.ucdavis.edu/math.PR/0603154 --------------------------------------------------------------- 4229. LARGE DEVIATIONS AND LAWS OF THE ITERATED LOGARITHM FOR THE LOCAL TIMES OF ADDITIVE STABLE PROCESSES Xia Chen We study the upper tail behaviors of the local times of the additive stable processes. Let $X_1(t),..., X_p(t)$ be independent, $d$-dimensional symmetric stable processes with stable index $0<\alpha\le 2$ and consider the additive stable process $\ol{X}(t_1,..., t_p)=X_1(t_1)+... +X_p(t_p)$. Under the condition $d<\alpha p$, we obtain a precise form of large deviation principle for the local time $$ \eta^x\big([0,t]^p\big)=\int_0^t...\int_0^t\delta_x\big(X_1(s_1)+... +X_p(s_p)\big)ds_1... ds_p $$ of the multi-parameter process $\ol{X} (t_1,..., t_p)$, and for its supremum norm $\displaystyle\sup_{x\in\R^d}\eta^x\big([0,t]^p\big)$. Our results apply to the law of the iterated logarithm and our approach is based on Fourier analysis, moment computation and time exponentiation. http://front.math.ucdavis.edu/math.PR/0603159 --------------------------------------------------------------- 4230. A DILUTED VERSION OF THE PERCEPTRON MODEL David Marquez-Carreras and Carles Rovira and Samy Tindel This note is concerned with a diluted version of the perceptron model. We establish a replica symmetric formula at high temperature, which is achieved by studying the asymptotic behavior of a given spin magnetization. Our main task will be to identify the order parameter of the system. http://front.math.ucdavis.edu/math.PR/0603162 --------------------------------------------------------------- 4231. JOINT SINGULAR VALUE DISTRIBUTION OF TWO CORRELATED RECTANGULAR GAUSSIAN MATRICES AND ITS APPLICATION Shuangquan Wang and Ali Abdi Let $\mathbf{H}=(h_{ij})$ and $\mathbf{G}=(g_{ij})$ be two $m\times n$, $m\leq n$, random matrices, each with i.i.d complex zero-mean unit- variance Gaussian entries, with correlation between any two elements given by $\mathbb{E}[h_{ij}g_{pq}^\star]=\rho \delta_{ip}\delta_{jq}$ such that $|\rho|<1$, where ${}^\star$ denotes the complex conjugate and $ \delta_{ij}$ is the Kronecker delta. Assume $\{s_k\}_{k=1}^m$ and $\{r_l\}_{l=1}^m$ are unordered singular values of $\mathbf{H}$ and $\mathbf{G}$, respectively, and $s$ and $r$ are randomly selected from $\{s_k\}_{k=1}^m$ and $\{r_l\}_ {l=1}^m$, respectively. In this paper, exact analytical closed-form expressions are derived for the joint probability distribution function (PDF) of $\{s_k\}_{k=1}^m$ and $\{r_l\}_{l=1}^m$ using an Itzykson-Zuber-type integral, as well as the joint marginal PDF of $s$ and $r$, by a bi-orthogonal polynomial technique. These PDFs are of interest in multiple-input multiple- output (MIMO) wireless communication channels and systems. http://front.math.ucdavis.edu/math.PR/0603170 --------------------------------------------------------------- 4232. FREEZING TRANSITION OF THE DIRECTED POLYMER IN A $1+D$ RANDOM MEDIUM : LOCATION OF THE CRITICAL TEMPERATURE AND UNUSUAL CRITICAL PROPERTIES Cecile Monthus and Thomas Garel In dimension $d \geq 3$, the directed polymer in a random medium undergoes a phase transition between a free phase and a disorder dominated phase. For the latter, Fisher and Huse have proposed a droplet theory based on the scaling of the free energy fluctuations $\Delta F(l) \sim l^{\theta}$. On the other hand, in related growth models belonging to the KPZ universality class, Forrest and Tang have found that the height-height correlation function is logarithmic at the transition. For the directed polymer model at criticality, this translates into logarithmic free energy fluctuations $\Delta F_{T_c}(l) \sim (\ln l)^{\sigma}$ with $\sigma=1/2$. In this paper, we propose a droplet scaling analysis exactly at criticality based on this logarithmic scaling. Our main conclusion is that the typical correlation length $\xi(T)$ of the low temperature phase, diverges as $ \ln \xi(T) \sim (- \ln (T_c-T))^{1/ \sigma} \sim (- \ln (T_c-T))^{2} $. Furthermore, the logarithmic dependence of $\Delta F_{T_c}(l)$ leads to the conclusion that the critical temperature $T_c$ actually coincides with the explicit upper bound $T_2$ derived by Derrida and coworkers, where $T_2$ corresponds to the temperature below which the ratio $\bar{Z_L^2}/(\bar{Z_L})^2$ diverges exponentially in $L$. Finally, since the Fisher-Huse droplet theory was initially introduced for the spin- glass phase, we briefly mention the similarities and differences with the directed polymer model. If one speculates that the free energy of droplet excitations for spin-glasses is also logarithmic at $T_c$, one obtains a logarithmic decay for the mean square correlation function at criticality $\bar{C^2(r)} \sim 1/(\ln r )^{\sigma}$. http://front.math.ucdavis.edu/cond-mat/0603041 --------------------------------------------------------------- 4233. ON THE 2D ISING WULFF CRYSTAL NEAR CRITICALITY Raphael Cerf and Reda Juerg Messikh We study the behavior of the two-dimensional Ising model in a finite box at temperatures that are below, but very close to, the critical temperature. In a regime where the temperature approaches the critical point and, simultaneously, the size of the box grows fast enough, we establish a large deviation principle that proves the appearance of a round Wulff crystal http://front.math.ucdavis.edu/math.PR/0603178 --------------------------------------------------------------- 4234. GAME-THEORETIC VERSIONS OF STRONG LAW OF LARGE NUMBERS FOR UNBOUNDED VARIABLES Masayuki Kumon and Akimichi Takemura and Kei Takeuchi We consider strong law of large numbers (SLLN) in the framework of game-theoretic probability of Shafer and Vovk (2001). We prove several versions of SLLN for the case that Reality's moves are unbounded. Our game- theoretic versions of SLLN largely correspond to standard measure-theoretic results. However game-theoretic proofs are different from measure-theoretic ones in the explicit consideration of various hedges. In measure-theoretic proofs existence of moments are assumed, whereas in our game-theoretic proofs we assume availability of various hedges to Skeptic for finite prices. http://front.math.ucdavis.edu/math.PR/0603184 --------------------------------------------------------------- 4235. ASYMPTOTICS FOR THE SMALL FRAGMENTS OF THE FRAGMENTATION AT NODES Romain Abraham (MAPMO) and Jean-Fran\c{c}ois Delmas (CERMICS) We consider the fragmentation at nodes of the L\'{e}vy continuous random tree introduced in a previous paper. In this framework we compute the asymptotic for the number of small fragments at time $\theta$. This limit is increasing in $\theta$ and discontinuous. In the $\alpha$-stable case the fragmentation is self-similar with index $1/\alpha$, with $\alpha \in (1,2)$ and the results are close to those Bertoin obtained for general self-similar fragmentations but with an additional assumtion which is not fulfilled here. http://front.math.ucdavis.edu/math.PR/0603192 --------------------------------------------------------------- 4236. FRAGMENTATION AT HEIGHT ASSOCIATED TO L\'{E}VY PROCESSES Jean-Fran\c{c}ois Delmas (CERMICS) We consider the height process of a L\'{e}vy process with no negative jumps, and its associated continuous tree representation. Using tools developed by Duquesne and Le Gall, we construct a fragmentation process at height, which generalizes the fragmentation at height of stable trees given by Miermont. In this more general framework, we recover that the dislocation measures are the same as the dislocation measures of the fragmentation at node introduced by Abraham and Delmas, up to a factor equal to the fragment size. We also compute the asymptotic for the number of small fragments. http://front.math.ucdavis.edu/math.PR/0603193 --------------------------------------------------------------- 4237. K-PROCESSES, SCALING LIMIT AND AGING FOR THE REM-LIKE TRAP MODEL Luiz Renato Fontes and Pierre Mathieu We study K-processes, which are Markov processes in a denumerable state space, all of whose elements are stable, with the exception of a single state, starting from which the process enters finite sets of stable states with uniform distribution. We show how these processes arise, in a particular instance, as scaling limits of the REM-like trap model ``at low temperature'', and subsequently derive aging results for those models in this context. http://front.math.ucdavis.edu/math.PR/0603198 --------------------------------------------------------------- 4238. SELF-SIMILARITY AND FRACTIONAL BROWNIAN MOTIONS ON LIE GROUPS F. Baudoin and L. Coutin The goal of this paper is to define and study a notion of fractional Brownian motion on a Lie group. We define it as at the solution of a stochastic differential equation driven by a linear fractional Brownian motion. We show that this process has stationary increments and satisfies a local self-similar property. Furthermore the Lie groups for which this self-similar property is global are characterized. Finally, we prove an integration by parts formula on the path group space and deduce the existence of a density. http://front.math.ucdavis.edu/math.PR/0603199 --------------------------------------------------------------- 4239. THE MAXIMUM OF A RANDOM WALK REFLECTED AT A GENERAL BARRIER Niels Richard Hansen We define the reflection of a random walk at a general barrier and derive, in case the increments are light tailed and have negative mean, a necessary and sufficient criterion for the global maximum of the reflected process to be finite a.s. If it is finite a.s., we show that the tail of the distribution of the global maximum decays exponentially fast and derive the precise rate of decay. Finally, we discuss an example from structural biology that motivated the interest in the reflection at a general barrier. http://front.math.ucdavis.edu/math.PR/0603208 --------------------------------------------------------------- 4240. ANALYSIS OF TOP TO BOTTOM-$K$ SHUFFLES Sharad Goel A deck of $n$ cards is shuffled by repeatedly moving the top card to one of the bottom $k_n$ positions uniformly at random. We give upper and lower bounds on the total variation mixing time for this shuffle as $k_n$ ranges from a constant to $n$. We also consider a symmetric variant of this shuffle in which at each step either the top card is randomly inserted into the bottom $k_n$ positions or a random card from the bottom $k_n$ positions is moved to the top. For this reversible shuffle we derive bounds on the $L^2$ mixing time. Finally, we transfer mixing time estimates for the above shuffles to the lazy top to bottom-$k$ walks that move with probability 1/2 at each step. http://front.math.ucdavis.edu/math.PR/0603209 --------------------------------------------------------------- 4241. OVERSHOOTS AND UNDERSHOOTS OF L\'{E}VY PROCESSES R. A. Doney and A. E. Kyprianou We obtain a new fluctuation identity for a general L\'{e}vy process giving a quintuple law describing the time of first passage, the time of the last maximum before first passage, the overshoot, the undershoot and the undershoot of the last maximum. With the help of this identity, we revisit the results of Kl\"{u}ppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004) 1766--1801] concerning asymptotic overshoot distribution of a particular class of L\'{e}vy processes with semi-heavy tails and refine some of their main conclusions. In particular, we explain how different types of first passage contribute to the form of the asymptotic overshoot distribution established in the aforementioned paper. Applications in insurance mathematics are noted with emphasis on the case that the underlying L\'{e}vy process is spectrally one sided. http://front.math.ucdavis.edu/math.PR/0603210 --------------------------------------------------------------- 4242. ESTIMATION OF ANISOTROPIC GAUSSIAN FIELDS THROUGH RADON TRANSFORM Hermine Bierm\'{e} (MAP5) We estimate the anisotropic index of an anisotropic fractional Brownian field. For all directions, we give a convergent estimator of the value of the anisotropic index in this direction, based on generalized quadratic variations. We also prove a central limit theorem. First we present a result of identification based on the asymptotic behavior of the spectral density of a process. Then, we define Radon transforms of the anisotropic fractional Brownian field and prove that these processes admit a spectral density satisfying the previous assumptions. http://front.math.ucdavis.edu/math.ST/0602663 --------------------------------------------------------------- 4243. ON A NONHIERARCHICAL VERSION OF THE GENERALIZED RANDOM ENERGY MODEL Erwin Bolthausen and Nicola Kistler We introduce a natural nonhierarchical version of Derrida's generalized random energy model. We prove that, in the thermodynamical limit, the free energy is the same as that of a suitably constructed GREM. http://front.math.ucdavis.edu/math.PR/0603212 --------------------------------------------------------------- 4244. A SCHEME FOR SIMULATING ONE-DIMENSIONAL DIFFUSION PROCESSES WITH DISCONTINUOUS COEFFICIENTS Antoine Lejay and Miguel Martinez The aim of this article is to provide a scheme for simulating diffusion processes evolving in one-dimensional discontinuous media. This scheme does not rely on smoothing the coefficients that appear in the infinitesimal generator of the diffusion processes, but uses instead an exact description of the behavior of their trajectories when they reach the points of discontinuity. This description is supplied with the local comparison of the trajectories of the diffusion processes with those of a skew Brownian motion. http://front.math.ucdavis.edu/math.PR/0603214 --------------------------------------------------------------- 4245. STOCHASTIC DYNAMICS OF DISCRETE CURVES AND EXCLUSION PROCESSES. PART 1: HYDRODYNAMIC LIMIT OF THE ASEP SYSTEM Guy Fayolle and Cyril Furtlehner This report is the foreword of a series dedicated to stochastic deformations of curves. Problems are set in terms of exclusion processes, the ultimate goal being to derive hydrodynamic limits for these systems after proper scalings. In this study, solely the basic \textsc{asep} system on the torus is analyzed. The usual sequence of empirical measures, converges in probability to a deterministic measure, which is the unique weak solution of a Cauchy problem. The method presents some new features, letting hope for extensions to higher dimension. It relies on the analysis of a family of parabolic differential operators, involving variational calculus. Namely, the variables are the values of functions at given points, their number being possibly infinite. http://front.math.ucdavis.edu/math.PR/0603215 --------------------------------------------------------------- 4246. CONTINUITY FOR SELF-DESTRUCTIVE PERCOLATION IN THE PLANE J. van den Berg and R. Brouwer and B. Vagvolgyi A few years ago two of us introduced, motivated by the study of certain forest-fireprocesses, the self-destructive percolation model (abbreviated as sdp model). A typical configuration for the sdp model with parameters p and delta is generated in three steps: First we generate a typical configuration for the ordinary percolation model with parameter p. Next, we make all sites in the infinite occupied cluster vacant. Finally, each site that was already vacant in the beginning or made vacant by the above action, becomes occupied with probability delta (independent of the other sites). Let theta(p, delta) be the probability that some specified vertex belongs, in the final configuration, to an infinite occupied cluster. In our earlier paper we stated the conjecture that, for the square lattice and other planar lattices, the function theta has a discontinuity at points of the form (p_c, delta), with delta sufficiently small. We also showed remarkable consequences for the forest-fire models. The conjecture naturally raises the question whether the function theta is continuous outside some region of the above mentioned form. We prove that this is indeed the case. An important ingredient in our proof is a (somewhat stronger form of a) recent ingenious RSW-like percolation result of Bollob\'{a}s and Riordan. http://front.math.ucdavis.edu/math.PR/0603223 --------------------------------------------------------------- 4247. ELEMENTS OF STOCHASTIC CALCULUS VIA REGULARISATION Francesco Russo and Pierre Vallois This paper first summarizes the foundations of stochastic calculus via regularization and constructs through this procedure It\^o and Stratonovich integrals. In the second part, a survey and new results are presented in relation with finite quadratic variation processes, Dirichlet and weak Dirichlet processes. http://front.math.ucdavis.edu/math.PR/0603224 --------------------------------------------------------------- 4248. ON THE CRITICAL BEHAVIOR AT THE LOWER PHASE TRANSITION OF THE CONTACT PROCESS Michael Aizenman and Paul Jung We present general results for the contact process by a method which applies to all transitive graphs of bounded degree, including graphs of exponential growth. The model's infection rates are varied though a common control parameter, for which two natural transition points are defined as: i. $\lambda_T$, the value up to which the infection dies out exponentially fast if introduced at a single site, and ii. $\lambda_H$, the threshold for the existence of an invariant measure with a non-vanishing density of infected sites. It is shown here that for all transitive graphs the two thresholds coincide. The method, which proceeds through partial differential inequalities for the infection density, yields also generally valid bounds on two related critical exponents. The work extends existing results whose derivations were restricted to either the discrete-time versions of the contact process or to graphs with subexponential growth. http://front.math.ucdavis.edu/math.PR/0603227 --------------------------------------------------------------- 4249. STRONG LOCALIZATION AND MACROSCOPIC ATOMS FOR DIRECTED POLYMERS Vincent Vargas (PMA) In this article, we derive strong localization results for directed polymers in random environment. We show that at "low temperature" the polymer measure is asymptotically concentrated at a few points of macroscopic mass (we call these points epsilon-atoms). These results are derived assuming weak conditions on the tail decay of the random environment. http://front.math.ucdavis.edu/math.PR/0603233 --------------------------------------------------------------- 4250. AN INVARIANCE PRINCIPLE FOR NEW WEAKLY DEPENDENT STATIONARY MODELS USING SHARP MOMENT ASSUMPTIONS Paul Doukhan (LS-CREST and SAMOS) and Olivier Wintenberger (SAMOS) This paper is aimed at sharpen a weak invariance principle for stationary sequences in Doukhan & Louhichi (1999). Our assumption is both beyond mixing and the causal $\theta$-weak dependence in Dedecker and Doukhan (2003); those authors obtained a sharp result which improves on an optimal one in Doukhan {\it et alii} (1995) under strong mixing. We prove this result and we also precise convergence rates under existence of moments with order $>2$ while Doukhan & Louhichi (1999) assume a moment of order $>4$. Analogously to those authors, we use a non-causal condition to deal with some general classes of stationary and weakly dependent sequences. Besides the previously used $\eta$- and $\kappa$-weak dependence conditions, we introduce a mixed condition, $\lambda$, adapted to consider Bernoulli shifts with dependent inputs. http://front.math.ucdavis.edu/math.ST/0603221 --------------------------------------------------------------- 4251. THRESHOLDS AND EXPECTATION THRESHOLDS Jeff Kahn and Gil Kalai Consider a random graph G in G(n,p) and the graph property: G contains a copy of a specific graph H. (An example to keep in mind: H is a Hamiltonian cycle.) Let p be the minimal value for which the expected number of copies of H' in G is at least 1/2 for every subgraph H' of H. Let q be the value for which the probability that G contains a copy of H is 1/2. Conjecture: q/p = O (log n). Related conjectures for general Boolean functions, and a possible connection with discrete isoperimetry are discussed. http://front.math.ucdavis.edu/math.CO/0603218 --------------------------------------------------------------- 4252. LARGE N LIMIT OF GAUSSIAN RANDOM MATRICES WITH EXTERNAL SOURCE, PART Pavel M. Bleher and Arno B.J. Kuijlaars We consider the double scaling limit in the random matrix ensemble with an external source $\frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM$ defined on $n\times n$ Hermitian matrices, where $A$ is a diagonal matrix with two eigenvalues $\pm a$ of equal multiplicities. The value $a=1$ is critical since the eigenvalues of $M$ accumulate as $n \to \infty$ on two intervals for $a > 1$ and on one interval for $0 < a < 1$. These two cases were treated in Parts I and II, where we showed that the local eigenvalue correlations have the universal limiting behavior known from unitary random matrix ensembles. For the critical case $a=1$ new limiting behavior occurs which is described in terms of Pearcey integrals, as shown by Br\'ezin and Hikami, and Tracy and Widom. We establish this result by applying the Deift/Zhou steepest descent method to a $3 \times 3$-matrix valued Riemann-Hilbert problem which involves the construction of a local parametrix out of Pearcey integrals. We resolve the main technical issue of matching the local Pearcey parametrix with a global outside parametrix by modifying an underlying Riemann surface. http://front.math.ucdavis.edu/math-ph/0602064 --------------------------------------------------------------- 4253. A FORWARD--BACKWARD STOCHASTIC ALGORITHM FOR QUASI-LINEAR PDES Fran\c{c}ois Delarue and St\'{e}phane Menozzi We propose a time-space discretization scheme for quasi-linear parabolic PDEs. The algorithm relies on the theory of fully coupled forward-- backward SDEs, which provides an efficient probabilistic representation of this type of equation. The derivated algorithm holds for strong solutions defined on any interval of arbitrary length. As a bypass product, we obtain a discretization procedure for the underlying FBSDE. In particular, our work provides an alternative to the method described in [Douglas, Ma and Protter (1996) Ann. Appl. Probab. 6 940--968] and weakens the regularity assumptions required in this reference. http://front.math.ucdavis.edu/math.PR/0603250 --------------------------------------------------------------- 4254. A NEW INVERSE FORMULA FOR THE LAPLAS'S TRANSFORMATION Andrey Pavlov In the article is proved,that the complex part of the analytical continuation of the LL(Z(x)) on the negative axis is equal to cZ(x),c=const., were Z(x) is the odd function from the wide class of functions,L(Z(x)) is the transformation of Laplas. http://front.math.ucdavis.edu/math.PR/0603258 From pas at www2.economia.unimi.it Fri Jul 28 00:32:25 2006 From: pas at www2.economia.unimi.it (pas@www2.economia.unimi.it) Date: Fri Jul 28 00:32:22 2006 Subject: [Pas] PAS has changed URL Message-ID: Dear PAS subscribers, due to a hardware change we needed to rename and move the URL of PAS on www2.economia.unimi.it (note the '2'). You don't need to do anything special. This is just a warning note. Next PAS letter will be sent on early august. the maintainer, Stefano M. Iacus ----------------------------------- Stefano M. Iacus Department of Economics, Business and Statistics University of Milan Via Conservatorio, 7 I-20123 Milan - Italy Ph.: +39 02 50321 461 Fax: +39 02 50321 505 http://www.economia.unimi.it/iacus ------------------------------------------------------------------------ ------------ Please don't send me Word or PowerPoint attachments if not absolutely necessary. See: http://www.gnu.org/philosophy/no-word-attachments.html From pas at www2.economia.unimi.it Tue Aug 1 10:02:33 2006 From: pas at www2.economia.unimi.it (pas@www2.economia.unimi.it) Date: Tue Aug 1 10:04:48 2006 Subject: [Pas] Probability Abstracts 93 Message-ID: <57DAA56A-B4E0-4F0E-9E54-B5495B3ED9AD@unimi.it> Aug 1st, 2006 Letter 93 Probability Abstract Service Abstracts from May-1-2006 to Jul-31-2006 html version here: http://www2.economia.unimi.it/PAS/Letters/ letter_93.shtml --------------------------------------------------------------- Note: this PAS letter cover three months instead of only two. This was due to a PAS server update during July 2006 which cause delay. Next PAS letter will have the same bimonthly posting. --------------------------------------------------------------- 4255. ESTIMATION IN SPIN GLASSES: A FIRST STEP Sourav Chatterjee The Sherrington-Kirkpatrick model of spin glasses, the Hopfield model of neural networks, and the Ising spin glass are all models of binary data belonging to the one-parameter exponential family with quadratic sufficient statistic. Under bare minimal conditions, we establish the consistency of the maximum pseudolikelihood estimate of the natural parameter in this family, even at critical temperatures. Since very little is known about the low and critical temperature regimes of these extremely difficult models, the proof requires several new ideas. The author's version of Stein's method is a particularly useful tool. One goal of this paper is to introduce these techniques into the realm of mathematical statistics through an example. http://front.math.ucdavis.edu/math.PR/0604634 --------------------------------------------------------------- 4256. A DELAYED BLACK AND SCHOLES FORMULA I Mercedes Arriojas and Yaozhong Hu and Salah-Eldin Mohammed and Gyula Pap In this article we develop an explicit formula for pricing European options when the underlying stock price follows a non-linear stochastic differential delay equation (sdde). We believe that the proposed model is sufficiently flexible to fit real market data, and is yet simple enough to allow for a closed-form representation of the option price. Furthermore, the model maintains the no-arbitrage property and the completeness of the market. The derivation of the option-pricing formula is based on an equivalent martingale measure. http://front.math.ucdavis.edu/math.PR/0604640 --------------------------------------------------------------- 4257. A DELAYED BLACK AND SCHOLES FORMULA II Mercedes Arriojas and Yaozhong Hu and Salah-Eldin Mohammed and Gyula Pap This article is a sequel to [A.H.M.P]. In [A.H.M.P], we develop an explicit formula for pricing European options when the underlying stock price follows a non-linear stochastic delay equation with fixed delays in the drift and diffusion terms. In this article, we look at models of the stock price described by stochastic functional differential equations with variable delays. We present a class of examples of stock dynamics with variable delays that permit an explicit form for the option pricing formula. As in [A.H.M.P], the market is complete with no arbitrage. This is achieved through the existence of an equivalent martingale measure. In subsequent work, the authors intend to test the models in [A.H.M.P] and the present article against real market data. http://front.math.ucdavis.edu/math.PR/0604641 --------------------------------------------------------------- 4258. THE HECKMAN-OPDAM MARKOV PROCESSES Bruno Schapira (MAPMO and PMA) We introduce and study the natural counterpart of the Dunkl Markov processes in a negatively curved setting. We give a semimartingale decomposition of the radial part, and some properties of the jumps. We prove also a law of large numbers, a central limit theorem, and the convergence of the normalized process to the Dunkl process. Eventually we describe the asymptotic behavior of the infinite loop as it was done by Anker, Bougerol and Jeulin in the symmetric spaces setting in \cite{ABJ}. http://front.math.ucdavis.edu/math.PR/0605020 --------------------------------------------------------------- 4259. TWO-DIMENSIONAL CRITICAL PERCOLATION: THE FULL SCALING LIMIT Federico Camia and Charles M. Newman We use SLE(6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -- that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved. http://front.math.ucdavis.edu/math.PR/0605035 --------------------------------------------------------------- 4260. GENERALIZATION OF THE BOREL-CANTELLI LEMMA Alexei Stepanov In the present note a generalization of Borel-Cantelli Lemma is proposed. http://front.math.ucdavis.edu/math.ST/0605007 --------------------------------------------------------------- 4261. TUG-OF-WAR AND THE INFINITY LAPLACIAN Yuval Peres and Oded Schramm and Scott Sheffield and David Wilson We prove that every bounded Lipschitz function F on a subset Y of a length space X admits a tautest extension to X, i.e., a unique Lipschitz extension u for which Lip_U u = Lip_{boundary of U} u for all open subsets U of X that do not intersect Y. This was previously known only for bounded domains R^n, in which case u is infinity harmonic, that is, a viscosity solution to Delta_infty u = 0. We also prove the first general uniqueness results for Delta_infty u = g on bounded subsets of R^n (when g is uniformly continuous and bounded away from zero), and analogous results for bounded length spaces. The proofs rely on a new game-theoretic description of u. Let u^epsilon(x) be the value of the following two-player zero-sum game, called tug-of- war: fix x_0=x \in X minus Y. At the kth turn, the players toss a coin and the winner chooses an x_k with d(x_k, x_{k-1})< \epsilon. The game ends when x_k is in Y, and player one's payoff is F(x_k) - (epsilon^2/2) sum_{i=0}^{k-1} g(x_i) We show that the u^\epsilon converge uniformly to u as epsilon tends to zero. Even for bounded domains in R^n, the game theoretic description of infinity-harmonic functions yields new intuition and estimates; for instance, we prove power law bounds for infinity-harmonic functions in the unit disk with boundary values supported in a delta-neighborhood of a Cantor set on the unit circle. http://front.math.ucdavis.edu/math.AP/0605002 --------------------------------------------------------------- 4262. OPERATORS ASSOCIATED WITH THE SOFT AND HARD SPECTRAL EDGES OF UNITARY ENSEMBLES Gordon Blower Using Hankel operators and shift-invariant subspaces on Hilbert space, this paper develops the theory of the operators associated with soft and hard edges of eigenvalue distributions of random matrices. Tracy and Widom introduced a projection operator $W$ to describe the soft edge of the spectrum of the Gaussian unitary ensemble. The subspace $WL^2$ is simply invariant under the translation semigroup $e^{itD}$ $(t\geq 0)$ and invariant under the Schr\"odinger semigroup $e^{it(D^2+x)}$ $(t\geq 0)$; these properties characterize $WL^2$ via Beurling's theorem. The Jacobi ensemble of random matrices has positive eigenvalues which tend to accumulate near to the hard edge at zero. This paper identifies a pair of unitary groups that satisfy the von Neumann--Weyl anti-commutation relations and leave invariant certain subspaces of $L^2(0,\infty)$ which are invariant for operators with Jacobi kernels. Such Tracy--Widom operators are reproducing kernels for weighted Hardy spaces, known as Sonine spaces. Periodic solutions of Hill's equation give a new family of Tracy--Widom type operators. http://front.math.ucdavis.edu/math.FA/0605010 --------------------------------------------------------------- 4263. A CENTRAL LIMIT THEOREM FOR CONVEX SETS B. Klartag We show that there exists a sequence $\eps_n \searrow 0$ for which the following holds: Let $K \subset \RR^n$ be a compact, convex set with a non-empty interior. Let $X$ be a random vector that is distributed uniformly in $K$. Then there exists a unit vector $\theta$ in $\RR^n$, $t_0 \in \RR $ and $\sigma > 0$ such that \begin{equation} \sup_{A \subset \RR} | Prob \{< X, \theta > \in A \} - \frac{1} {\sqrt{2 \pi \sigma}} \int_A e^{-\frac{(t - t_0)^2}{2 \sigma^2}} dt | \leq \eps_n, \end{equation} where the supremum runs over all measurable sets $A \subset \RR$, and where $<\cdot, \cdot >$ denotes the usual scalar product in $\RR^n$. Moreover, under the additional assumptions that the expectation of $X $ is zero and that the covariance matrix of $X$ is the identity matrix, we argue that most unit vectors $\theta$ satisfy ($\dagger$), with $t_0 = 0$ and $ \sigma = 1$. Thus, typical one-dimensional marginal distributions of high- dimensional, isotropic, convex sets are approximately gaussian. This proves a basic conjecture in asymptotic convex geometry, that was put forward by Anttila, Ball and Perissinaki and by Brehm and Voigt. We also discuss normal approximation for multi-dimensional marginal distributions of uniform measures on convex sets. http://front.math.ucdavis.edu/math.MG/0605014 --------------------------------------------------------------- 4264. PRICING WITH COHERENT RISK Alexander S. Cherny This paper deals with applications of coherent risk measures to pricing in incomplete markets. Namely, we study the No Good Deals pricing technique based on coherent risk. Two forms of this technique are presented: one defines a good deal as a trade with negative risk; the other one defines a good deal as a trade with unusually high RAROC. For each technique, the fundamental theorem of asset pricing and the form of the fair price interval are presented. The model considered includes static as well as dynamic models, models with an infinite number of assets, models with transaction costs, and models with portfolio constraints. In particular, we prove that in a model with proportional transaction costs the fair price interval converges to the fair price interval in a frictionless model as the coefficient of transaction costs tends to zero. Moreover, we study some problems in the ``pure'' theory of risk measures: we present a simple geometric solution of the capital allocation problem and apply it to define the coherent risk contribution. The mathematical tools employed are probability theory, functional analysis, and finite-dimensional convex analysis. http://front.math.ucdavis.edu/math.PR/0605049 --------------------------------------------------------------- 4265. ON THE RANGE OF THE SIMPLE RANDOM WALK BRIDGE ON GROUPS Itai Benjamini and Roey Izkovsky and Harry Kesten Let G be a vertex transitive graph. A study of the range of simple random walk on G and of its bridge is proposed. While it is expected that on a graph of polynomial growth the sizes of the range of the unrestricted random walk and of its bridge are the same in first order, this is not the case on some larger graphs such as regular trees. Of particular interest is the case when G is the Cayley graph of a group. In this case we even study the range of a general symmetric (not necessarily simple) random walk on G. We hope that the few examples for which we calculate the first order behavior of the range here will help to discover some relation between the group structure and the behavior of the range. Further problems regarding bridges are presented. http://front.math.ucdavis.edu/math.PR/0605050 --------------------------------------------------------------- 4266. EQUILIBRIUM WITH COHERENT RISK Alexander S. Cherny This paper is the continuation of "Pricing with coherent risk" and deals with further applications of coherent risk measures to problems of finance. First, we study the optimization problem. Three forms of this problem are considered. Furthermore, the results obtained are applied to the optimality pricing. Again three forms of this technique are considered. Finally, we study the equilibrium problem both in the unconstrained and in the constrained forms. We establish the equivalence between the global and the competitive optima and give a dual description of the equilibrium. Moreover, we provide an explicit geometric solution of the constrained equilibrium problem. Most of the results are presented on two levels: on a general level the results have a probabilistic form; for a static model with a finite number of assets, the results have a geometric form. http://front.math.ucdavis.edu/math.PR/0605051 --------------------------------------------------------------- 4267. LARGE DEVIATIONS AND A KRAMERS' TYPE LAW FOR SELF-STABILIZING DIFFUSIONS Samuel Herrmann and Peter Imkeller and Dierk Peithmann We investigate exit times from domains of attraction for the motion of a self-stabilized particle travelling in a geometric (potential type) landscape and perturbed by Brownian noise of small amplitude. Self- stabilization is mediated by an ensemble-average attraction adding on to the individual potential drift, where the particle is supposed to be suspended in a large population of identical ones. A Kramers' type law for the particle's exit from the potential's domains of attraction and a large deviations principle for the self-stabilizing diffusion are proved. It turns out that the exit law for the self-stabilizing diffusion coincides with the exit law of a potential diffusion without self-stabilization with a drift component perturbed by average attraction. We show that self-stabilization may substantially delay the exit from domains of attraction, and that the exit location may be completely different. http://front.math.ucdavis.edu/math.PR/0605053 --------------------------------------------------------------- 4268. OPTIMAL STOPPING OF HUNT AND L\'EVY PROCESSES Ernesto Mordecki and Paavo Salminen The optimal stopping problem for a Hunt processes on $\R$ is considered via the representation theory of excessive functions. In particular, we focus on infinite horizon (or perpetual) problems with one-sided structure, that is, there exists a point $x^*$ such that the stopping region is of the form $[x^*,+\infty)$. Corresponding results for two-sided problems are also indicated. The main result is a spectral representation of the value function in terms of the Green kernel of the process. Specializing in L\'evy processes, we obtain, by applying the Wiener-Hopf factorization, a general representation of the value function in terms of the maximum of the L\'evy process. To illustrate the results, an explicit expression for the Green kernel of Brownian motion with exponential jumps is computed and some optimal stopping problems for Poisson process with positive exponential jumps and negative drift are solved. http://front.math.ucdavis.edu/math.PR/0605054 --------------------------------------------------------------- 4269. SUR LE NOMBRE DE POINTS VISIT\'{E}S PAR UNE MARCHE AL\'{E} ATOIRE SUR UN AMAS INFINI DE PERCOLATION Clement Rau (LATP) In this article, we consider random walk on the infinite cluster of bond percolation on $\Z^d (d \geq 2)$. We show that the Laplace transformation of the number of visited points $N\_n$, has a behaviour as the random walk was on $\Z^d$. More precisely, for all $0<\alpha<1$, we proved that there exist constants $C\_i$ and $C\_s$ such that for all infinite cluster that contains the origin, we have: $$ e^{-C\_i n^{\frac{d}{d+2}}} \leq \E\_0^{\omega} (\alpha^{N\_n}) \leq e^{-C\_sn^{\frac{d}{d+2}}}.$$ Our approach is based on finding an isoperimetric inequalities on the infinite cluster, lifted on a wreath product which give good behaviour. The problem of the isoperimetry on wreath product was already raised by A.Ershler. http://front.math.ucdavis.edu/math.PR/0605056 --------------------------------------------------------------- 4270. COHERENT MEASUREMENT OF FACTOR RISKS Alexander S. Cherny and Dilip B. Madan We propose a new procedure for the risk measurement of large portfolios. It employs the following objects as the building blocks: - coherent risk measures introduced by Artzner, Delbaen, Eber, and Heath; - factor risk measures introduced in this paper, which assess the risks driven by particular factors like the price of oil, S&P500 index, or the credit spread; - risk contributions and factor risk contributions, which provide a coherent alternative to the sensitivity coefficients. We also propose two particular classes of coherent risk measures called Alpha V@R and Beta V@R, for which all the objects described above admit an extremely simple empirical estimation procedure. This procedure uses no model assumptions on the structure of the price evolution. Moreover, we consider the problem of the risk management on a firm's level. It is shown that if the risk limits are imposed on the risk contributions of the desks to the overall risk of the firm (rather than on their outstanding risks) and the desks are allowed to trade these limits within a firm, then the desks automatically find the globally optimal portfolio. http://front.math.ucdavis.edu/math.PR/0605062 --------------------------------------------------------------- 4271. PRICING AND HEDGING IN INCOMPLETE MARKETS WITH COHERENT RISK Alexander S. Cherny and Dilip B. Madan We propose a pricing technique based on coherent risk measures, which enables one to get finer price intervals than in the No Good Deals pricing. The main idea consists in splitting a liability into several parts and selling these parts to different agents. The technique is closely connected with the convolution of coherent risk measures and equilibrium considerations. Furthermore, we propose a way to apply the above technique to the coherent estimation of the Greeks. http://front.math.ucdavis.edu/math.PR/0605064 --------------------------------------------------------------- 4272. CAPM, REWARDS, AND EMPIRICAL ASSET PRICING WITH COHERENT RISK Alexander S. Cherny and Dilip B. Madan The paper has 2 main goals: 1. We propose a variant of the CAPM based on coherent risk. 2. In addition to the real-world measure and the risk- neutral measure, we propose the third one: the extreme measure. The introduction of this measure provides a powerful tool for investigating the relation between the first two measures. In particular, this gives us - a new way of measuring reward; - a new approach to the empirical asset pricing. http://front.math.ucdavis.edu/math.PR/0605065 --------------------------------------------------------------- 4273. ITO MAPS AND ANALYSIS ON PATH SPACES K. D. Elworthy and Xue-Mei Li We consider versions of Malliavin calculus on path spaces of compact manifolds with diffusion measures, defining Gross-Sobolev spaces of differentiable functions and proving their intertwining with solution maps, I, of certain stochastic differential equations. This is shown to shed light on fundamental uniqueness questions for this calculus including uniqueness of the closed derivative operator $d$ and Markov uniqueness of the associated Dirichlet form. A continuity result for the divergence operator by Kree and Kree is extended to this situation. The regularity of conditional expectations of smooth functionals of classical Wiener space, given I, is considered and shown to have strong implications for these questions. A major role is played by the (possibly sub-Riemannian) connections induced by stochastic differential equations: Damped Markovian connections are used for the covariant derivatives. http://front.math.ucdavis.edu/math.PR/0605089 --------------------------------------------------------------- 4274. COMPRESSING REDUNDANT INFORMATION IN MARKOV CHAINS Giacomo Aletti Given a strongly stationary Markov chain and a finite set of stopping rules, we prove the existence of a polynomial algorithm which projects the Markov chain onto a minimal Markov chain without redundant information. Markov complexity is hence defined and tested on some classical problems. http://front.math.ucdavis.edu/math.PR/0605099 --------------------------------------------------------------- 4275. EXPECTED NUMBER OF LOCAL MAXIMA OF SOME GAUSSIAN RANDOM POLYNOMIALS S. Shemehsavar and S. Rezakhah Let $Q_n(x)=\sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraic polynomial where the coefficients $A_0,A_1,... $ form a sequence of centered Gaussian random variables. Moreover, assume that the increments $\Delta_j=A_j-A_{j-1}$, $j=0,1,2,...$ are independent, $A_{-1}=0$. The coefficients can be considered as $n$ consecutive observations of a Brownian motion. We study the asymptotic behaviour of the expected number of local maxima of $Q_n(x)$ below level $u=O(n^k)$, for some $k>0$. http://front.math.ucdavis.edu/math.PR/0605116 --------------------------------------------------------------- 4276. ANCHORED CRITICAL PERCOLATION CLUSTERS AND 2-D ELECTROSTATICS P. Kleban and J. J. H. Simmons and and R. M. Ziff We consider the densities of clusters, at the percolation point of a two-dimensional system, which are anchored in various ways to an edge. These quantities are calculated by use of conformal field theory and computer simulations. We find that they are given by simple functions of the potentials of 2-D electrostatic dipoles, and that a kind of superposition {\it cum} factorization applies. Our results broaden this connection, already known from previous studies, and we present evidence that it is more generally valid. An exact result similar to the Kirkwood superposition approximation emerges. http://front.math.ucdavis.edu/cond-mat/0605120 --------------------------------------------------------------- 4277. THE CONFIGURATIONAL MEASURE ON MUTUALLY AVOIDING SLE PATHS Michael J. Kozdron (University of Regina) and Gregory F. Lawler (Cornell University) We define multiple chordal SLEs in a simply connected domain by considering a natural configurational measure on paths. We show how to construct these measures so that they are conformally covariant and satisfy certain boundary perturbation and Markov properties, as well as a cascade relation. As an example of our construction, we derive the scaling limit of Fomin's identity in the case of two paths directly; that is, we prove that the probability that an SLE(2) and a Brownian excursion do not intersect can be given in terms of the determinant of the excursion hitting matrix. Finally, we define the lambda-SAW, a one-parameter family of measures on self-avoiding walks on Z^2. http://front.math.ucdavis.edu/math.PR/0605159 --------------------------------------------------------------- 4278. LOOP-FREE MARKOV CHAINS AS DETERMINANTAL POINT PROCESSES Alexei Borodin We show that any loop-free Markov chain on a discrete space can be viewed as a determinantal point process. As an application we prove central limit theorems for the number of particles in a window for renewal processes and Markov renewal processes with Bernoulli noise. http://front.math.ucdavis.edu/math.PR/0605168 --------------------------------------------------------------- 4279. BEHAVIOR OF A SECOND CLASS PARTICLE IN HAMMERSLEY'S PROCESS Eric Cator and Sergei Dobrynin In the case of a rarefaction fan in a non-stationary Hammersley process, we explicitly calculate the asymptotic behavior of the process as we move out along a ray, and the asymptotic distribution of the angle within the rarefaction fan of a second class particle and a dual second class particle. Furthermore, we consider a stationary Hammersley process and use the previous results to show that trajectories of a second class particle and a dual second class particles touch with probability one, and we give some information on the area enclosed by the two trajectories, up until the first intersection point. This is linked to the area of influence of an added Poisson point in the plane. http://front.math.ucdavis.edu/math.PR/0605199 --------------------------------------------------------------- 4280. RANDOM MATRIX CENTRAL LIMIT THEOREMS FOR NON-INTERSECTING RANDOM WALKS Jinho Baik and Toufic Suidan We consider non-intersecting random walks satisfying the condition that the increments have a finite moment generating function. We prove that in a certain limiting regime where the number of walks and the number of time steps grow to infinity, several limiting distributions of the walks at the mid-time behave as the eigenvalues of random Hermitian matrices as the dimension of the matrices grows to infinity. http://front.math.ucdavis.edu/math.PR/0605212 --------------------------------------------------------------- 4281. ON THE BEHAVIOR OF RANDOM WALK AROUND HEAVY POINTS Endre Cs\'aki and Ant\'onia F\"oldes and P\'al R\'ev\'esz Consider a symmetric aperiodic random walk in $Z^d$, $d\geq 3$. There are points (called heavy points) where the number of visits by the random walk is close to its maximum. We investigate the local times around these heavy points and show that they converge to a deterministic limit as the number of steps tends to infinity. http://front.math.ucdavis.edu/math.PR/0605221 --------------------------------------------------------------- 4282. $T^{1/3}$ SUPERDIFFUSIVITY OF FINITE-RANGE ASYMMETRIC EXCLUSION PROCESSES ON $\MATHBB Z$ Jeremy Quastel and Benedek Valko We consider finite-range asymmetric exclusion processes on $\mathbb Z $ with non-zero drift. The diffusivity $D(t)$ is expected to be of $O(t^ {1/3})$. We prove that $D(t)\ge Ct^{1/3}$ in the weak (Tauberian) sense that $ \int_0^\infty e^{-\lambda t}tD(t)dt \ge C\lambda^{-7/3}$ as $\lambda\to 0$. The proof employs the resolvent method to make a direct comparison with the totally asymmetric simple exclusion process, for which the result is a consequence of the scaling limit for the two-point function recently obtained by Ferrari and Spohn. When $p(z)\ge p(-z)$ for each $z>0$, we show further that $tD(t)$ is monotone, and hence we can conclude that $D(t)\ge Ct^{1/3}(\log t)^{-7/3}$ in the usual sense. http://front.math.ucdavis.edu/math.PR/0605266 --------------------------------------------------------------- 4283. THE MULTIPARAMETER FRACTIONAL BROWNIAN MOTION Erick Herbin and Ely Merzbach We define and study the multiparameter fractional Brownian motion. This process is a generalization of both the classical fractional Brownian motion and the multiparameter Brownian motion, when the condition of independence is relaxed. Relations with the L\'evy fractional Brownian motion and with the fractional Brownian sheet are discussed. Different notions of stationarity of the increments for a multiparameter process are studied and applied to the fractional property. Using self-similarity we present a characterization for such processes. Finally, behavior of the multiparameter fractional Brownian motion along increasing paths is analysed. http://front.math.ucdavis.edu/math.PR/0605279 --------------------------------------------------------------- 4284. MULTISERVER QUEUEING SYSTEMS WITH RETRIALS AND ABANDONMENTS AND THEIR APPLICATION TO CALL CENTERS Vyacheslav M. Abramov The paper studies multiserver retrial queueing systems with $m$ servers. Arrival process is a quite general point process. An arriving customer occupies one of free servers. If upon arrival all servers are busy, then the customer waits for his service in orbit, and after random time retries more and more to occupy a server. The orbit has one waiting space only, and arriving customer, who finds all servers busy and the waiting space occupied, abandons the system. Time intervals between possible retrials are assumed to have arbitrary distribution (the retrial scheme is exactly explained in the paper). The paper provides analysis of this system. Specifically the paper studies optimal number of servers to decrease the loss proportion to a given value. The representation obtained for loss proportion enables us to solve the problem numerically. The algorithm for numerical solution includes effective simulation, which meets the challenge of rare events problem in simulation. Application of the results to call centers is discussed as well. http://front.math.ucdavis.edu/math.PR/0605285 --------------------------------------------------------------- 4285. A LIMIT THEOREM FOR THE MAXIMAL INTERPOINT DISTANCE OF A RANDOM SAMPLE IN THE UNIT BALL Michael Mayer and Ilya Molchanov We prove a limit theorem for the the maximal interpoint distance (also called the diameter) for a sample of n i.i.d. points in the unit ball of dimension 2 or more. The exact form of the limit distribution and the required normalisation are derived using assumptions on the tail of the interpoint distance for two i.i.d. points. The results are specialised for the cases when the points have spherical symmetric distributions, in particular, are uniformly distributed in the whole ball and on its boundary. http://front.math.ucdavis.edu/math.PR/0605289 --------------------------------------------------------------- 4286. CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD Oded Schramm and Scott Sheffield We prove that the chordal contour lines of the discrete Gaussian free field converge to forms of SLE(4). Specifically, there is a constant lambda > 0 such that when h is an interpolation of the discrete Gaussian free field on a Jordan domain -- with boundary values -lambda on one boundary arc and lambda on the complementary arc -- the zero level line of h joining the endpoints of these arcs converges to SLE(4) as the domain grows larger. If instead the boundary values are -a < 0 on the first arc and b > 0 on the complementary arc, then the convergence is to SLE(4;a/lambda-1,b/lambda-1), a variant of SLE(4). http://front.math.ucdavis.edu/math.PR/0605337 --------------------------------------------------------------- 4287. TOWARD THE BEST CONSTANT FACTOR FOR THE RADEMACHER-GAUSSIAN TAIL COMPARISON Iosif Pinelis Let S_n:=a_1\vp_1+...+a_n\vp_n, where \vp_1,...,\vp_n are independent Rademacher random variables (r.v.'s) and a_1,...,a_n are any real numbers such that a_1^2+...+a_n^2=1. Let Z be a standard normal r.v. It is proved that the best constant factor c in inequality \P(S_n>x) \leq c\P(Z>x) for all x in \R is between two explicitly defined absolute constants c_1 and c_2 such that c_10, we prove that the Green functions are comparable, provided D is connected. These results apply for example to alpha-stable relativistic process. This process was studied in recent years. In the paper we also considered one dimensional case for alpha<= 1 and proved that the Green functions for an open and bounded interval are comparable. http://front.math.ucdavis.edu/math.PR/0605370 --------------------------------------------------------------- 4292. POISSON APPROXIMATIONS FOR THE ISING MODEL David Coupier A $d$-dimensional Ising model on a lattice torus is considered. As the size $n$ of the lattice tends to infinity, a Poisson approximation is given for the distribution of the number of copies in the lattice of any given local configuration, provided the magnetic field $a=a(n)$ tends to $-\infty $ and the pair potential $b$ remains fixed. Using the Stein-Chen method, a bound is given for the total variation error in the ferromagnetic case. http://front.math.ucdavis.edu/math.PR/0605395 --------------------------------------------------------------- 4293. AN EXPLICIT BOUND ON THE LOGARITHMIC SOBOLEV CONSTANT OF WEAKLY DEPENDENT RANDOM VARIABLES Katalin Marton We prove logarithmic Sobolev inequality for measures $$ q^n(x^n)=\text{dist}(X^n)=\exp\bigl(-V(x^n)\bigr), \quad x^n\in \Bbb R^n, $$ under the assumptions that: (i) the conditional distributions $$ Q_i (\cdot| x_j, j\neq i)=\text{dist}(X_i| X_j= x_j, j\neq i) $$ satisfy a logarithmic Sobolev inequality with a common constant $\rho$, and (ii) they also satisfy some condition expressing that the mixed partial derivatives of the Hamiltonian $V$ are not too large relative to $\rho$. \bigskip Condition (ii) has the form that the norms of some matrices defined in terms of the mixed partial derivatives of $V$ do not exceed $1/2\cdot\rho\cdot(1-\de)$. The logarithmic Sobolev constant of $q^n$ can then be estimated from below by $1/2\cdot\rho\cdot\delta$. This improves on earlier results by Th. Bodineau and B. Helffer, by giving an explicit bound, for the logarithmic Sobolev constant for $q^n$. http://front.math.ucdavis.edu/math.PR/0605397 --------------------------------------------------------------- 4294. POISSON LIMITS FOR EMPIRICAL POINT PROCESSES Andr\'{e} Dabrowski and Gail Ivanoof and Rafal Kulik Define the scaled empirical point process on an independent and identically distributed sequence $\{Y_i: i\le n\}$ as the random point measure with masses at $a_n^{-1} Y_i$. For suitable $a_n$ we obtain the weak limit of these point processes through a novel use of a dimension-free method based on the convergence of compensators of multiparameter martingales. The method extends previous results in several directions. We obtain limits at points where the density of $Y_i$ may be zero, but has regular variation. The joint limit of the empirical process evaluated at distinct points is given by independent Poisson processes. These results also hold for multivariate $Y_i$ with little additional effort. Applications are provided both to nearest- neighbour density estimation in high dimensions, and to the asymptotic behaviour of multivariate extremes such as those arising from bivariate normal copulas. http://front.math.ucdavis.edu/math.PR/0605400 --------------------------------------------------------------- 4295. DECAY PROPERTIES OF THE CONNECTIVITY FOR MIXED LONG RANGE PERCOLATION MODELS ON $\Z^D$ Gastao A. Braga and Leandro M. Cioletti and Remy Sanchis In this paper we consider mixed short-long range independent bond percolation models on $\Z^d$. Let $p_{uv}$ be the probability that the edge $(u,v) $ will be open. Successive applications of the Simon-Lieb inequality at a fixed length scale generates convolutions of $p_{uv}$ with itself which yields, in the perturbative regime, that the long distance behavior of the connectivity $\tau_{xy}$ is governed by the probability $p_{xy}$. Allowing a $x,y$- dependent length scale and using a multi-scale analysis due to Aizenman and Newman, decay properties of $\tau_{xy}$ are obtained up to the critical point. http://front.math.ucdavis.edu/math-ph/0605047 --------------------------------------------------------------- 4296. UNIVERSALITY FOR THE DISTANCE IN FINITE VARIANCE RANDOM GRAPHS: EXTENDED VERSION Henri van den Esker and Remco van der Hofstad and Gerard Hooghiemstra The asymptotic behavior of the graph distance between two uniformly chosen nodes in the configuration model is generalized to a wide class of random graphs, where the degrees have finite variance. Among others, this class contains the Poissonian random graph and the generalized random graph (including the classical Erd\H{o}s-R\'enyi graph). We prove that the graph distance grows like $\log_\nu N$, when the base of the logarithm equals $\nu = E[\Lambda^2]/E[\Lambda]$, where $\Lambda$ is a positive random variable with $P(\Lambda> x)\leq cx^{1-\tau}$, for some constant $c$ and some power-law exponent $\tau>3$. In addition, the random fluctuations around this asymptotic mean $\log_\nu N$ are characterized and shown to be uniformly bounded. The proof of this result uses that the graph distance of all members of the class can be coupled successfully to the graph distance in the Poissonian random graph. http://front.math.ucdavis.edu/math.PR/0605414 --------------------------------------------------------------- 4297. SMALL DEVIATIONS OF GAUSSIAN RANDOM FIELDS IN $L_Q$--SPACES Mikhail Lifshits and Werner Linde and Zhan Shi We investigate small deviation properties of Gaussian random fields in the space $L_q(\R^N,\mu)$ where $\mu$ is an arbitrary finite compactly supported Borel measure. Of special interest are hereby "thin" measures $\mu$, i.e., those which are singular with respect to the $N$--dimensional Lebesgue measure; the so--called self--similar measures providing a class of typical examples. For a large class of random fields (including, among others, fractional Brownian motions), we describe the behavior of small deviation probabilities via numerical characteristics of $\mu$, called mixed entropy, characterizing size and regularity of $\mu$. For the particularly interesting case of self--similar measures $ \mu$, the asymptotic behavior of the mixed entropy is evaluated explicitly. As a consequence, we get the asymptotic of the small deviation for $N$-- parameter fractional Brownian motions with respect to $L_q(\R^N,\mu)$--norms. While the upper estimates for the small deviation probabilities are proved by purely probabilistic methods, the lower bounds are established by analytic tools concerning Kolmogorov and entropy numbers of H\"older operators. http://front.math.ucdavis.edu/math.PR/0605417 --------------------------------------------------------------- 4298. IMBALANCE ATTRACTORS FOR A STRATEGIC MODEL OF MARKET MICROSTRUCTURE Ted Theodosopoulos and Ming Yuen In this paper we extend the series of our studies on the properties of an interacting particle model for market microstructure. In our earlier work we defined a Markov process on the majority opinion of the agents, obtained the transition probabilities and analyzed the martingale properties of the ensuing wealth process. Here we relax the assumption on the choices of individual agents by allowing mixed strategies, offering opportunities for the agents to gain intermediate submartingale exposure for their individual wealth processes. We develop a novel two-dimensional spin system to model the critical regions of the wealth process as a reflection of the agents' behaviors. We exhibit strategic conflicts between individual market participants and the market as a whole, and identify a new source of uncertainty arising from `reinforced expectations'. http://front.math.ucdavis.edu/math.PR/0605421 --------------------------------------------------------------- 4299. GENERALIZED 3G THEOREM AND APPLICATION TO RELATIVISTIC STABLE PROCESS ON NON-SMOOTH OPEN SETS Panki Kim and Young-Ran Lee Let G(x,y) and G_D(x,y) be the Green functions of rotationally invariant symmetric \alpha-stable process in R^d and in an open set D respectively, where 0<\alpha < 2. The inequality G_D(x,y)G_D(y,z)/G_D(x,z) \le c(G(x,y)+G (y,z)) is a very useful tool in studying (local) Schrodinger operators. When the above inequality is true with a constant c=c(D)>0, then we say that the 3G theorem holds in D. In this paper, we establish a generalized version of 3G theorem when D is a bounded \kappa-fat open set, which includes a bounded John domain. The 3G we consider is of the form G_D(x,y)G_D(z,w)/G_D(x,w), where y may be different from z. When y=z, we recover the usual 3G. The 3G form G_D(x,y)G_D(z,w)/G_D(x,w) appears in non-local Schrodinger operator theory. Using our generalized 3G theorem, we give a concrete class of functions belonging to the non-local Kato class, introduced by Chen and Song, on \kappa-fat open sets. As an application, we discuss relativistic \alpha-stable processes (relativistic Hamiltonian when \alpha=1) in \kappa-fat open sets. We identify the Martin boundary and the minimal Martin boundary with the Euclidean boundary for relativistic \alpha-stable processes in \kappa-fat open sets. Furthermore, we show that relative Fatou type theorem is true for relativistic stable processes in \kappa-fat open sets. The main results of this paper hold for a large class of symmetric Markov processes, as are illustrated in the last section of this paper. We also discuss the generalized 3G theorem for a large class of symmetric stable Levy processes. http://front.math.ucdavis.edu/math.PR/0605422 --------------------------------------------------------------- 4300. SUFFICIENT CONDITIONS FOR THE INVERTIBILITY OF ADAPTED PERTURBATIONS OF IDENTITY ON THE WIENER SPACE Ali Suleyman Ustunel and Moshe Zakai Let $(W,H,\mu)$ be the classical Wiener space. Assume that $U=I_W+u$ is an adapted perturbation of identity, i.e., $u:W\to H$ is adapted to the canonical filtration of $W$. We give some sufficient analytic conditions on $u$ which imply the invertibility of the map $U$. In particular it is shown that if $u\in \DD_{p,1}(H)$ is adapted and if $\exp({1/2}\|\nabla u\|_2^2-\delta u)\in L^q(\mu)$, where $p^{-1}+q^{-1}=1$, then $I_W+u$ is almost surely invertible. As a consequence, if, there exists an integer $k\geq 1$ such that $\| \nabla^k u\|_{H^{\otimes(k+1)}}\in L^\infty(\mu)$, then $I_W+u$ is again almost surely invertible. http://front.math.ucdavis.edu/math.PR/0605433 --------------------------------------------------------------- 4301. RESAMPLING FROM THE PAST TO IMPROVE ON MCMC ALGORITHMS Yves F. Atchade We introduce the idea that resampling from past observations in a Markov Chain Monte Carlo sampler can fasten convergence. We prove that proper resampling from the past does not disturb the limit distribution of the algorithm. We illustrate the method with two examples. The first on a Bayesian analysis of stochastic volatility models and the other on Bayesian phylogeny reconstruction. http://front.math.ucdavis.edu/math.ST/0605452 --------------------------------------------------------------- 4302. INFINITELY DIVISIBILITY OF SOLUTIONS OF SOME SEMI-STABLE INTEGRO-DIFFERENTIAL EQUATIONS AND EXPONENTIAL FUNCTIONALS OF LEVY PROCESSES Pierre Patie We provide the increasing $q$-harmonic functions associated to spectrally negative semi-stable Feller semigroups, which have been introduced by Lamperti. The functions are expressed in terms of a new family of power series which includes, for instance, the modified Bessel functions of the first kind and some new generalization of the Mittag-Leffler function. Then, we show that some specific combinations of these functions are Laplace transforms of selfdecomposable or infinitely divisible distributions concentrated on the positive line. In particular, this generalizes the result of Hartman in the case of the Bessel semigroup. Finally, when the Levy process has a negative mean, we compute the associated decreasing $q$-harmonic functions and derive the Laplace transform of the exponential functionals. http://front.math.ucdavis.edu/math.PR/0605453 --------------------------------------------------------------- 4303. HYBRID DYNAMICS FOR CURRENCY MODELING Ted Theodosopoulos and Alex Trifunovic We present a simple hybrid dynamical model as a tool to investigate behavioral strategies based on trend following. The multiplicative symbolic dynamics are generated using a lognormal diffusion model for the at- the-money implied volatility term structure. Thus, are model exploits information from derivative markets to obtain qualititative properties of the return distribution for the underlier. We apply our model to the JPY-USD exchange rate and the corresponding 1mo., 3mo., 6mo. and 1yr. implied volatilities. Our results indicate that the modulation of autoregressive trend following using derivative-based signals significantly improves the fit to the distribution of times between successive sign flips in the underlier time series. http://front.math.ucdavis.edu/math.PR/0605457 --------------------------------------------------------------- 4304. ON STABLE PARETO LAWS IN A HIERARCHICAL MODEL OF ECONOMY Alexander M. Chebotarev This study considers a model of the income distribution of agents whose pairwise interaction is asymmetric and price-invariant. Asymmetric transactions are typical for chain-trading groups who arrange their business such that commodities move from senior to junior partners and money moves in the opposite direction. The price-invariance of transactions means that the probability of a pairwise interaction is a function of the ratio of incomes, which is independent of the price scale or absolute income level. These two features characterize the hierarchical model. The income distribution in this class of models is a well-defined double-Pareto function, which possesses Pareto tails for the upper and lower incomes. For gross and net upper incomes, the model predicts definite values of the Pareto exponents, $a_{\rm gross}$ and $a_{\rm net}$, which are stable with respect to quantitative variation of the pair-interaction. The Pareto exponents are also stable with respect to the choice of a demand function within two classes of status-dependent behavior of agents: linear demand ($a_{\rm gross}=1$, $a_{\rm net}=2$) and unlimited slowly varying demand ($a_{\rm gross}=a_{\rm net}=1$). For the sigmoidal demand that describes limited returns, $a_{\rm gross}=a_{\rm net}=1+\alpha$, with some $\alpha>0$ satisfying a transcendental equation. The low-income distribution may be singular or vanishing in the neighborhood of the minimal income; in any case, it is $L_1$-integrable and its Pareto exponent is given explicitly. The theory used in the present study is based on a simple balance equation and new results from multiplicative Markov chains and exponential moments of random geometric progressions. http://front.math.ucdavis.edu/math.PR/0605461 --------------------------------------------------------------- 4305. STABILITY OF PROCESSOR SHARING NETWORKS WITH SIMULTANEOUS RESOURCE REQUIREMENTS Jennie Hansen and Cian Reynolds and Stan Zachary We study the phenomenon of entrainment in processor sharing networks, whereby, while individual network resources have sufficient capacity to met demand, the requirement for simultaneous availability of resources means that a network may nevertheless be unstable. We show that instability occurs through poor control, and that, for a variety of network topologies, only small modifications to controls are required in order to ensure stability. For controls which possess a natural monotonicity property, we give some new results for the classification of the corresponding Markov processes, which lead to conditions both for stability and for instability. http://front.math.ucdavis.edu/math.PR/0605477 --------------------------------------------------------------- 4306. ON THE OCCUPATION MEASURE OF SUPER-BROWNIAN MOTION J.F. Le Gall and M. Merle We derive the asymptotic behavior of the occupation measure of the unit ball, for super-Brownian motion started from the Dirac measure at a distant point x and conditioned to hit the unit ball. In the critical dimension d=4, we obtain a limiting exponential distribution for the ratio of the occupation measure over log(|x|). http://front.math.ucdavis.edu/math.PR/0605482 --------------------------------------------------------------- 4307. RANDOM REAL TREES J.F. Le Gall We survey recent developments about random real trees, whose prototype is the Continuum Random Tree (CRT) introduced by Aldous in 1991. We briefly explain the formalism of real trees, which yields a neat presentation of the theory and in particular of the relations between discrete Galton-Watson trees and continuous random trees. We then discuss the particular class of self- similar random real trees called stable trees, which generalize the CRT. We review several important results concerning stable trees, including their branching property, which is analogous to the well-known property of Galton- Watson trees, and the calculation of their fractal dimension. We then consider spatial trees, which combine the genealogical structure of a real tree with spatial displacements, and we explain their connections with superprocesses. In the last section, we deal with a particular conditioning problem for spatial trees, which is closely related to asymptotics for random planar quadrangulations. http://front.math.ucdavis.edu/math.PR/0605484 --------------------------------------------------------------- 4308. AN ALGEBRAIC APPROACH OF POLYA PROCESSES Nicolas Pouyanne (LM-Versailles) P\'olya processes are natural generalization of P\'olya-Eggenberger urn models. This article presents a new approach of their asymptotic behaviour {\it via} moments, based on the spectral decomposition of a suitable finite difference operator on polynomial functions. Especially, it provides new results for {\it large} processes (a P\'olya process is called {\it small} when 1 is simple eigenvalue of its replacement matrix and when any other eigenvalue has a real part $\leq 1/2$; otherwise, it is called large). http://front.math.ucdavis.edu/math.CO/0605472 --------------------------------------------------------------- 4309. ON THE LIKELIHOOD OF COMPARABILITY IN BRUHAT ORDER Adam Hammett and Boris Pittel The poset of permutations of [n] under Bruhat ordering is studied. We give nontrivial upper and lower bounds for the number of comparable pairs of permutations in both the weak and strong versions of this order. In light of numerical experiments, we conjecture that in either case the upper bound is qualitatively close to the actual number of comparable pairs. http://front.math.ucdavis.edu/math.PR/0605490 --------------------------------------------------------------- 4310. LARGE DEVIATIONS FOR WEIGHTED EMPIRICAL MEAN WITH OUTLIERS Myl\`ene Ma\"{\i}da and Jamal Najim and Sandrine P\'ech\'e We study in this article large deviations for the empirical mean of iid random vectors with some deterministic weights, whose empirical measure weakly converges to some compactly support probability distribution. The scope of this paper is to study the effect on the LDP of outliers, that is sequences of weights that remain far from the support of the limiting measure. http://front.math.ucdavis.edu/math.PR/0605491 --------------------------------------------------------------- 4311. ZERO-ONE LAWS FOR BINARY RANDOM FIELDS David Coupier and Paul Doukhan and Bernard Ycart A set of binary random variables indexed by a lattice torus is considered. Under a mixing hypothesis, the probability of any proposition belonging to the first order logic of colored graphs tends to 0 or 1, as the size of the lattice tends to infinity. For the particular case of the Ising model with bounded pair potential and surface potential tending to $-\infty$, the threshold functions of local propositions are computed, and sufficient conditions for the zero-one law are given. http://front.math.ucdavis.edu/math.PR/0605502 --------------------------------------------------------------- 4312. ON CLASSES OF NON-GAUSSIAN ASYMPTOTIC MINIMIZERS IN ENTROPIC UNCERTAINTY PRINCIPLES S. Zozor and C. Vignat In this paper we revisit the Bialynicki-Birula & Mycielski uncertainty principle and its cases of equality. This Shannon entropic version of the well-known Heisenberg uncertainty principle can be used when dealing with variables that admit no variance. In this paper, we extend this uncertainty principle to Renyi entropies. We recall that in both Shannon and Renyi cases, and for a given dimension n, the only case of equality occurs for Gaussian random vectors. We show that as n grows, however, the bound is also asymptotically attained in the cases of n-dimensional Student-t and Student-r distributions. A complete analytical study is performed in a special case of a Student-t distribution. We also show numerically that this effect exists for the particular case of a n-dimensional Cauchy variable, whatever the Renyi entropy considered, extending the results of Abe and illustrating the analytical asymptotic study of the student-t case. In the Student-r case, we show numerically that the same behavior occurs for uniformly distributed vectors. These particular cases and other ones investigated in this paper are interesting since they show that this asymptotic behavior cannot be considered as a "Gaussianization" of the vector when the dimension increases. http://front.math.ucdavis.edu/math.PR/0605510 --------------------------------------------------------------- 4313. PHASE TRANSITIONS IN A PIECEWISE EXPANDING COUPLED MAP LATTICE WITH LINEAR NEAREST NEIGHBOUR COUPLING Jean-Baptiste Bardet (IRMAR) and Gerhard Keller We construct a mixing continuous piecewise linear map on [-1,1] with the property that a two-dimensional lattice made of these maps with a linear north and east nearest neighbour coupling admits a phase transition. We also provide a modification of this construction where the local map is an expanding analytic circle map. The basic strategy is borroughed from [Gielis- MacKay (2000)], namely we compare the dynamics of the CML to those of a probabilistic cellular automaton of Toom's type. http://front.math.ucdavis.edu/math.DS/0605501 --------------------------------------------------------------- 4314. POTENTIAL THEORY OF TRUNCATED STABLE PROCESSES Panki Kim and Renming Song For any 0 < alpha <2, a truncated symmetric alpha-stable process is a symmetric Levy process in R^d with a Levy density given by c|x|^{-d- alpha} 1_{|x|< 1} for some constant c. In this paper we study the potential theory of truncated symmetric stable processes in detail. We prove a Harnack inequality for nonnegative harmonic nonnegative functions these processes. We also establish a boundary Harnack principle for nonnegative functions which are harmonic with respect to these processes in bounded convex domains. We give an example of a non-convex domain for which the boundary Harnack principle fails. http://front.math.ucdavis.edu/math.PR/0605533 --------------------------------------------------------------- 4315. EXPONENTIAL APPROXIMATION BY EXCHANGEABLE PAIRS AND SPECTRAL GRAPH THEORY Sourav Chatterjee and Jason Fulman A general Berry-Esseen bound is obtained for the exponential distribution using Stein's method of exchangeable pairs. As an application, an error term is derived for Hora's result that the spectrum of the Bernoulli-Laplace Markov chain has an exponential limit. This is the first use of Stein's method to study the spectrum of a graph with a non-normal limit. http://front.math.ucdavis.edu/math.PR/0605552 --------------------------------------------------------------- 4316. ON DUAL PROCESSES OF NON-SYMMETRIC DIFFUSIONS WITH MEASURE- VALUED DRIFTS Panki Kim and Renming Song In this paper, we study properties of the dual process and Schrodinger-type operators of a non-symmetric diffusion with measure-valued drift. Let mu=(mu^1,..., mu^d) be such that each mu^i is a signed measure on R^d belonging to the Kato class K_{d, 1}. We show that a killed diffusion process with measure-valued drift in any bounded domain has a dual process with respect to a certain reference measure. For an arbitrary bounded domain, we show that a scale invariant Harnack inequality is true for the dual process. We also show that, if the domain is bounded C^{1,1}, the boundary Harnack principle for the dual process is true and the (minimal) Martin boundary for the dual process can be identified with the Euclidean boundary. It is also shown that the harmonic measure for the dual process is locally comparable to that of the h- conditioned Brownian motion with h being the ground state. Under the gaugeability assumption, if the domain is bounded Lipschitz, the (minimal) Martin boundary for the Schrodinger operator obtained from the diffusion with measure- value drift can be identified with the Euclidean boundary. http://front.math.ucdavis.edu/math.PR/0605556 --------------------------------------------------------------- 4317. ESTIMATES ON GREEN FUNCTIONS AND SCHRODINGER-TYPE EQUATIONS FOR NON-SYMMETRIC DIFFUSIONS WITH MEASURE-VALUED DRIFTS Panki Kim and Renming Song In this paper, we establish sharp two-sided estimates for the Green functions of non-symmetric diffusions with measure-valued drifts in bounded Lipschitz domains. As consequences of these estimates, we get a 3G type theorem and a conditional gauge theorem for these diffusions in bounded Lipschitz domains. We also establish two-sided estimates for the heat kernels of Schrodinger-type operators with measure-valued potential in bounded C^{1,1}-domains and a scale invariant boundary Harnack principle for the positive harmonic functions with respect to Schrodinger-type operators in bounded Lipschitz domains. http://front.math.ucdavis.edu/math.PR/0605557 --------------------------------------------------------------- 4318. ON TAYLOR DISPERSION IN OSCILLATORY CHANNEL FLOWS Kalvis M. Jansons We revisit Taylor dispersion in oscillatory flows at zero Reynolds number, giving an alternative method of calculating the Taylor dispersivity that is easier to use with computer algebra packages to obtain exact expressions. We consider the effect of out-of-phase oscillatory shear and Poiseuille flow, and show that the resulting Taylor dispersivity is independent of the phase difference. We also determine exact expressions for several examples of oscillatory power-law fluid flows. http://front.math.ucdavis.edu/math.PR/0605561 --------------------------------------------------------------- 4319. PARTITION FUNCTION OF PERIODIC ISORADIAL DIMER MODELS B\'eatrice de Tili\`ere Isoradial dimer models were introduced in \cite{Kenyon3} - they consist of dimer models whose underlying graph satisfies a simple geometric condition, and whose weight function is chosen accordingly. In this paper, we prove a conjecture of \cite{Kenyon3}, namely that for periodic isoradial dimer models, the growth rate of the toroidal partition function has a simple explicit formula involving the local geometry of the graph only. This is a surprising feature of periodic isoradial dimer models, which does not hold in the general periodic dimer case \cite{KOS}. http://front.math.ucdavis.edu/math.PR/0605583 --------------------------------------------------------------- 4320. MODELLING DERIVATIVES PRICING MECHANISMS WITH THEIR GENERATING FUNCTIONS Shige Peng In this paper we study dynamic pricing mechanisms of financial derivatives. A typical model of such pricing mechanism is the so-called g-- expectation defined by solutions of a backward stochastic differential equation with g as its generating function. Black-Scholes pricing model is a special linear case of this pricing mechanism. We are mainly concerned with two types of pricing mechanisms in an option market: the market pricing mechanism through which the market prices of options are produced, and the ask-bid pricing mechanism operated through the system of market makers. The later one is a typical nonlinear pricing mechanism. Data of prices produced by these two pricing mechanisms are usually quoted in an option market. We introduce a criteria, i.e., the domination condition (A5) in (2.5) to test if a dynamic pricing mechanism under investigation is a g--pricing mechanism. This domination condition was statistically tested using CME data documents. The result of test is significantly positive. We also provide some useful characterizations of a pricing mechanism by its generating function. http://front.math.ucdavis.edu/math.PR/0605599 --------------------------------------------------------------- 4321. LARGE DEVIATIONS FOR SUMS DEFINED ON A GALTON-WATSON PROCESS Klaus Fleischmann and Vitali Wachtel In this paper we study the large deviation behavior of sums of i.i.d. random variables X_i defined on a supercritical Galton-Watson process Z. We assume the finiteness of the moments EX_1^2 and EZ_1log Z_1. The underlying interplay of the partial sums of the X_i and the lower deviation probabilities of Z is clarified. Here we heavily use lower deviation probability results on Z we recently published in [FW06]. http://front.math.ucdavis.edu/math.PR/0605617 --------------------------------------------------------------- 4322. SPATIAL BIRTH AND DEATH PROCESSES AS SOLUTIONS OF STOCHASTIC EQUATIONS Nancy L. Garcia and Thomas G. Kurtz Spatial birth and death processes are obtained as solutions of a system of stochastic equations. The processes are required to be locally finite, but may involve an infinite population over the full (noncompact) type space. Conditions are given for existence and uniqueness of such solutions, and for temporal and spatial ergodicity. For birth and death processes with constant death rate, a sub-criticality condition on the birth rate implies that the process is ergodic and converges exponentially fast to the stationary distribution. http://front.math.ucdavis.edu/math.PR/0605620 --------------------------------------------------------------- 4323. THE LARGEST EIGENVALUE OF RANK ONE DEFORMATION OF LARGE WIGNER MATRICES Delphine F\'eral and Sandrine P\'ech\'e The purpose of this paper is to establish universality of the fluctuations of the largest eigenvalue of some non necessarily Gaussian complex Deformed Wigner Ensembles. The real model is also considered. Our approach is close to the one used by A. Soshnikov in the investigations of classical real or complex Wigner Ensembles. It is based on the computation of moments of traces of high powers of the random matrices under consideration. http://front.math.ucdavis.edu/math.PR/0605624 --------------------------------------------------------------- 4324. ON THE MAXIMUM QUEUE LENGTH IN THE SUPERMARKET MODEL Malwina J. Luczak and Colin McDiarmid There are $n$ queues, each with a single server. Customers arrive in a Poisson process at rate $\lambda n$, where $0<\lambda<1$. Upon arrival each customer selects $d\geq2$ servers uniformly at random, and joins the queue at a least-loaded server among those chosen. Service times are independent exponentially distributed random variables with mean 1. We show that the system is rapidly mixing, and then investigate the maximum length of a queue in the equilibrium distribution. We prove that with probability tending to 1 as $n\to\infty$ the maximum queue length takes at most two values, which are $\ln\ln n/\ln d+O(1)$. http://front.math.ucdavis.edu/math.PR/0605639 --------------------------------------------------------------- 4325. THE SIZE OF COMPONENTS IN CONTINUUM NEAREST-NEIGHBOR GRAPHS Iva Kozakova and Ronald Meester and Seema Nanda We study the size of connected components of random nearest-neighbor graphs with vertex set the points of a homogeneous Poisson point process in ${\mathbb{R}}^d$. The connectivity function is shown to decay superexponentially, and we identify the exact exponent. From this we also obtain the decay rate of the maximal number of points of a path through the origin. We define the generation number of a point in a component and establish its asymptotic distribution as the dimension $d$ tends to infinity. http://front.math.ucdavis.edu/math.PR/0605640 --------------------------------------------------------------- 4326. DYNAMICAL STABILITY OF PERCOLATION FOR SOME INTERACTING PARTICLE SYSTEMS AND $\EPSILON$-MOVABILITY Erik I. Broman and Jeffrey E. Steif In this paper we will investigate dynamic stability of percolation for the stochastic Ising model and the contact process. We also introduce the notion of downward and upward $\epsilon$-movability which will be a key tool for our analysis. http://front.math.ucdavis.edu/math.PR/0605641 --------------------------------------------------------------- 4327. MONOTONICITY, ASYMPTOTIC NORMALITY AND VERTEX DEGREES IN RANDOM GRAPHS Svante Janson We exploit a result by Nerman which shows that conditional limit theorems hold when a certain monotonicity condition is satisfied. Our main result is an application to vertex degrees in random graphs where we obtain asymptotic normality for the number of vertices with a given degree in the random graph G(n,m) with a fixed number of edges from the corresponding result for the random graph G(n,m) with independent edges. We give also some simple applications to random allocations and to spacings. Finally, inspired by these results but logically independent from them, we investigate whether a one-sided version of the Cramer-Wold theorem holds. We show that such a version holds under a weak supplementary condition, but not without it. http://front.math.ucdavis.edu/math.PR/0605642 --------------------------------------------------------------- 4328. COMPARISON OF WEIGHTED AND UNWEIGHTED HISTOGRAMS N.D. Gagunashvili Two modifications of the chi square test for comparing usual (unweighted) and weighted histograms and two weighted histograms are proposed. Numerical examples illustrate an application of the tests for the histograms with different statistics of events. Proposed tests can be used for the comparison of experimental data histograms against simulated data histograms and two simulated data histograms. http://front.math.ucdavis.edu/physics/0605123 --------------------------------------------------------------- 4329. INTERMITTENCY ON CATALYSTS: SYMMETRIC EXCLUSION J. Gaertner and F. den Hollander and G. Maillard We continue our study of intermittency for the parabolic Anderson equation $\partial u/\partial t = \kappa\Delta u + \xi u$, where $u\colon \Z^d \times [0,\infty)\to\R$, $\kappa$ is the diffusion constant, $\Delta$ is the discrete Laplacian, and $\xi\colon \Z^d\times [0,\infty)\to\R$ is a space-time random medium. The solution of the equation describes the evolution of a ``reactant'' $u$ under the influence of a ``catalyst'' $\xi$. In this paper we focus on the case where $\xi$ is exclusion with a symmetric random walk transition kernel, starting from equilibrium with density $\rho\in (0,1)$. We consider the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of $u$. We show that these exponents are trivial when the random walk is recurrent, but display an interesting dependence on the diffusion constant $\kappa$ when the random walk is transient, with qualitatively different behavior in different dimensions. Special attention is given to the asymptotics of the exponents for $\kappa\to\infty$, which is controlled by moderate deviations of $\xi$ requiring a delicate expansion argument. In G\"artner and den Hollander \cite{garhol04} the case where $\xi $ is a Poisson field of independent (simple) random walks was studied. The two cases show interesting differences and similarities. Throughout the paper, a comparison of the two cases plays a crucial role. http://front.math.ucdavis.edu/math.PR/0605657 --------------------------------------------------------------- 4330. A VERSION OF H\"ORMANDER'S THEOREM FOR THE FRACTIONAL BROWNIAN MOTION F. Baudoin and M. Hairer It is shown that the law of an SDE driven by fractional Brownian motion with Hurst parameter greater than 1/2 has a smooth density with respect to Lebesgue measure, provided that the driving vector fields satisfy H\"ormander's condition. The main new ingredient of the proof is an extension of Norris' lemma to this situation. http://front.math.ucdavis.edu/math.PR/0605658 --------------------------------------------------------------- 4331. QUASI STATIONARY DISTRIBUTIONS AND FLEMING-VIOT PROCESSES IN COUNTABLE SPACES Pablo A. Ferrari and Nevena Maric We consider an irreducible pure jump Markov process with rates Q=(q (x,y)) on \Lambda\cup\{0\} with \Lambda countable and 0 an absorbing state. A quasi-stationary distribution (qsd) is a probability measure \nu on \Lambda that satisfies: starting with \nu, the conditional distribution at time t, given that at time t the process has not been absorbed, is still \nu. That is, \nu(x) = \nu P_t(x)/(\sum_{y\in\Lambda}\nu P_t(y)), with P_t the transition probabilities for the process with rates Q. A Fleming-Viot (fv) process is a system of N particles moving in \Lambda. Each particle moves independently with rates Q until it hits the absorbing state 0; but then instantaneously chooses one of the N-1 particles remaining in \Lambda and jumps to its position. Between absorptions each particle moves with rates Q independently. Under the condition \alpha:=\sum_x\inf Q(\cdot,x) > \sup Q(\cdot, 0):=C we prove existence of qsd for Q; uniqueness has been proven by Jacka and Roberts. When \alpha>0 the {\fv} process is ergodic for each N. Under \alpha>C the mean normalized densities of the fv unique stationary measure converge to the qsd of Q, as N \to \infty; in this limit the variances vanish. http://front.math.ucdavis.edu/math.PR/0605665 --------------------------------------------------------------- 4332. ON THE AVERAGE NUMBER OF SHARP CROSSINGS OF CERTAIN GAUSSIAN RANDOM POLYNOMIALS S. Shemehsavar and S. Rezakhah Let $Q_n(x)=\sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraic polynomial where the coefficients $A_0,A_1,... $ form a sequence of centered Gaussian random variables. Moreover, assume that the increments $\Delta_j=A_j-A_{j-1}$, $j=0,1,2,...$ are independent, assuming $A_{-1}=0$. The coefficients can be considered as $n$ consecutive observations of a Brownian motion. We obtain the asymptotic behaviour of the expected number of u-sharp crossings of polynomial $Q_n(x)$ . We refer to u-sharp crossings as those zero up-crossings with slope greater than $u$, or those down-crossings with slope smaller than $-u $. We consider the cases where $u$ is unbounded and is increasing with $n$, where $u=o(n^{5/4})$, and $u=o(n^{3/2})$ separately. http://front.math.ucdavis.edu/math.PR/0605699 --------------------------------------------------------------- 4333. ASYMPTOTIC BEHAVIOUR OF THE SIMPLE RANDOM WALK ON THE 2-COMB Daniela Bertacchi We analyze the differences between the horizontal and the vertical component of the simple random walk on the 2-dimensional comb. In particular we evaluate by combinatorial methods the asymptotic behaviour of the expected value of the distance from the origin, the maximal deviation and the maximal span in $n$ steps, proving that for all these quantities the order is $n^{1/4}$ for the horizontal projection and $n^{1/2}$ for the vertical one (the exact constants are determined). Then we rescale the two projections of the random walk dividing by $n^{1/4}$ and $n^{1/2}$ the horizontal and vertical ones, respectively. The limit process is obtained. As a corollary of the estimate of the expected value of the maximal deviation, the walk dimension is determined, showing that the Einstein relation between the fractal, spectral and walk dimensions does not hold on the comb. http://front.math.ucdavis.edu/math.PR/0605718 --------------------------------------------------------------- 4334. DIGITAL SEARCH TREES AND CHAOS GAME REPRESENTATION Peggy C\'{e}nac (INRIA Rocquencourt) and Brigitte Chauvin (LM- Versailles), St\'{e}phane Ginouillac (LM-Versailles), Nicolas Pouyanne (LM-Versailles) In this paper, we consider a possible representation of a DNA sequence in a quaternary tree, in which on can visualize repetitions of subwords. The CGR-tree turns a sequence of letters into a digital search tree (DST), obtained from the suffixes of the reversed sequence. Several results are known concerning the height and the insertion depth for DST built from i.i.d. successive sequences. Here, the successive inserted wors are strongly dependent. We give the asymptotic behaviour of the insertion depth and of the length of branches for the CGR-tree obtained from the suffixes of reversed i.i.d. or Markovian sequence. This behaviour turns out to be at first order the same one as in the case of independent words. As a by-product, asymptotic results on the length of longest runs in a Markovian sequence are obtained. http://front.math.ucdavis.edu/math.PR/0605719 --------------------------------------------------------------- 4335. ON THE BROWNIAN MEANDER AND EXCURSION CONDITIONED TO HAVE A FIXED TIME AVERAGE Lorenzo Zambotti We study the density of the time average of the Brownian meander/ excursion over the time interval [0,1]. Moreover we give an expression for the Brownian meander/excursion conditioned to have a fixed time average. http://front.math.ucdavis.edu/math.PR/0605720 --------------------------------------------------------------- 4336. INTRINSIC ULTRACONTRACTIVITY OF NON-SYMMETRIC DIFFUSIONS WITH MEASURE-VALUED DRIFTS AND POTENTIALS Panki Kim and Renming Song Recently we extended the concept of intrinsic ultracontractivity to non-symmetric semigroups. In this paper, we study the intrinsic ultracontractivity of non-symmetric diffusions with measure-valued drifts and measure-valued potentials in bounded domains. We show that scale invariant parabolic and elliptic Harnack inequalities are valid for this process. In this paper, we prove the parabolic boundary Harnack principle and the intrinsic ultracontractivity for the killed diffusion with measure- valued drift and potential when the domain is one of the following types of bounded domains: twisted Holder domains of order (1/3, 1], uniformly Holder domains of order (0, 2) and domains which can be locally represented as the region above the graph of a function. As a consequence of the intrinsic ultracontractivity, we get that the supremum of the expected conditional lifetimes finite. http://front.math.ucdavis.edu/math.PR/0605757 --------------------------------------------------------------- 4337. ZEROS OF RANDOM POLYNOMIALS ON C^M Thomas Bloom and Bernard Shiffman For a regular compact set $K$ in $C^m$ and a measure $\mu$ on $K$ satisfying the Bernstein-Markov inequality, we consider the ensemble $P_N$ of polynomials of degree $N$, endowed with the Gaussian probability measure induced by $L^2(\mu)$. We show that for large $N$, the simultaneous zeros of $m$ polynomials in $P_N$ tend to concentrate around the Silov boundary of $K$; more precisely, their expected distribution is asymptotic to $N^m \mu_{eq} $, where $\mu_{eq}$ is the equilibrium measure of $K$. For the case where $K$ is the unit ball, we give scaling asymptotics for the expected distribution of zeros as $N\to\infty$. http://front.math.ucdavis.edu/math.CV/0605739 --------------------------------------------------------------- 4338. THE OSTROGRADSKY SERIES AND RELATED PROBABILITY MEASURES S.Albeverio and O.Baranovskyi and M.Pratsiovytyi and G.Torbin We develop a metric and probabilistic theory for the Ostrogradsky representation of real numbers, i.e., the expansion of a real number $x$ in the following form: \begin{align*} x&= \sum_n\frac{(-1)^{n-1}}{q_1q_2... q_n}= &=\sum_n\frac{(-1)^{n-1}}{g_1(g_1+g_2)...(g_1+g_2+...+g_n)}\equiv \bO1(g_1,g_2,...,g_n,...), \end{align*} where $q_{n+1}>q_n\in\N$, $g_1=q_1$, $g_{k+1}=q_{k+1}-q_k$. We compare this representation with the corresponding one in terms of continued fractions. We establish basic metric relations (equalities and inequalities for ratios of the length of cylindrical sets). We also compute the Lebesgue measure of subsets belonging to some classes of closed nowhere dense sets defined by characteristic properties of the $\bO1$-representation. In particular, the conditions for the set $\Cset{V}$, consisting of real numbers whose $\bO1$-symbols take values from the set $V \subset N$, to be of zero resp. positive Lebesgue measure are found. For a random variable $\xi$ with independent $\bO1$-symbols $g_n(\xi)$ we prove the theorem establishing the purity of the distribution. In the case of singularity the conditions for such distributions to be of Cantor type are also found. http://front.math.ucdavis.edu/math.NT/0605747 --------------------------------------------------------------- 4339. SINGULAR PROBABILITY DISTRIBUTIONS AND FRACTAL PROPERTIES OF SETS OF REAL NUMBERS DEFINED BY THE ASYMPTOTIC FREQUENCIES OF THEIR S-ADIC DIGITS S.Albeverio and M.Pratsiovytyi and G.Torbin Properties of the set $T_s$ of "particularly non-normal numbers" of the unit interval are studied in details ($T_s$ consists of real numbers $x$, some of whose s-adic digits have the asymptotic frequencies in the nonterminating $s-$ adic expansion of $x$, and some do not). It is proven that the set $T_s$ is residual in the topological sense (i.e., it is of the first Baire category) and it is generic in the sense of fractal geometry ($T_s$ is a superfractal set, i.e., its Hausdorff-Besicovitch dimension is equal to~1). A topological and fractal classification of sets of real numbers via analysis of asymptotic frequencies of digits in their s-adic expansions is presented. http://front.math.ucdavis.edu/math.NT/0605763 --------------------------------------------------------------- 4340. SIMPLE TRANSIENT RANDOM WALKS IN ONE-DIMENSIONAL RANDOM ENVIRONMENT: THE CENTRAL LIMIT THEOREM I. Ya. Goldsheid We consider a simple random walk (dimension one, nearest neighbour jumps) in a quenched random environment. The goal of this work is to provide sufficient conditions, stated in terms of properties of the environment, under which the Central Limit Theorem (CLT) holds for the position of the walk. Verifying these conditions leads to a complete solution of the problem in the case of independent identically distributed environments as well as in the case of uniformly ergodic (and thus also weakly mixing) environments. http://front.math.ucdavis.edu/math.PR/0605775 --------------------------------------------------------------- 4341. OPTIMAL CONTROL FOR ROUGH DIFFERENTIAL EQUATIONS Laurent Mazliak (PMA) and Ivan Nourdin (PMA) In this note, we consider an optimal control problem associated to a differential equation driven by a H\"{o}lder continuous function g of index greater than 1/2. We split our study in two cases. If the coefficient of dg\_t does not depend on the control process, we prove an existence theorem for a slightly generalized control problem, that is we obtain a literal extension of the corresponding deterministic situation. If the coefficient of dg \_t depends on the control process, we also prove an existence theorem but we are here obliged to restrict the set of controls to sufficiently regular functions. http://front.math.ucdavis.edu/math.PR/0606030 --------------------------------------------------------------- 4342. SHUFFLING CARDS FOR BLACKJACK, BRIDGE, AND OTHER CARD GAMES Mark Conger and D. Viswanath This paper is about the following question: How many riffle shuffles mix a deck of card for games such as blackjack and bridge? An object that comes up in answering this question is the descent polynomial associated with pairs of decks, where the decks are allowed to have repeated cards. We prove that the problem of computing the descent polynomial given a pair of decks is $#P$-complete. We also prove that the coefficients of these polynomials can be approximated using the bell curve. However, as must be expected in view of the $#P$-completeness result, approximations using the bell curve are not good enough to answer our question. Some of our answers to the main question are supported by theorems, and others are based on experiments supported by heuristic arguments. In the introduction, we carefully discuss the validity of our answers. http://front.math.ucdavis.edu/math.PR/0606031 --------------------------------------------------------------- 4343. LONG-TIME BEHAVIOR OF STOCHASTIC MODEL WITH MULTI-PARTICLE SYNCHRONIZATION Anatoly Manita We consider a basic stochastic particle system consisting of $N$ identical particles with isotropic $k$-particle synchronization, $k\geq 2$. In the limit when both number of particles $N$ and time $t=t(N)$ grow to infinity we study an asymptotic behavior of a coordinate spread of the particle system. We describe three time stages of $t(N)$ for which a qualitative behavior of the system is completely different. Moreover, we discuss the case when a spread of the initial configuration depends on $N$ and increases to infinity as $N\to \infty $. http://front.math.ucdavis.edu/math.PR/0606040 --------------------------------------------------------------- 4344. SIEVING AND THE ERD{\H O}S-KAC THEOREM Andrew Granville and K. Soundararajan We give a relatively easy proof of the Erd\H os-Kac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results in the literature. http://front.math.ucdavis.edu/math.NT/0606039 --------------------------------------------------------------- 4345. THE POISSON BOUNDARY OF LAMPLIGHTER RANDOM WALKS ON TREES Anders Karlsson and Wolfgang Woess Let T be the homogeneous tree with degree and G a finitely generated group whose Cayley graph is T. The associated lamplighter group is the wreath product of the cyclic group of order r with G. For a large class of random walks on this group, we prove almost sure convergence to a natural geometric boundary. If the probability law governing the random walk has finite first moment, then the probability space formed by this geometric boundary together with the limit distribution of the random walk is proved to be maximal, that is, the Poisson boundary. We also prove that the Dirichlet problem at infinity is solvable for continuous functions on the active part of the boundary, if the lamplighter "operates at bounded range". http://front.math.ucdavis.edu/math.PR/0606046 --------------------------------------------------------------- 4346. RECURRENCE AND TRANSIENCE FOR BRANCHING RANDOM WALKS IN AN IID RANDOM ENVIRONMENT Sebastian M\"uller We give three different criteria for transience of a Branching Markov Chain. These conditions enable us to give a classification of Branching Random Walks in Random Environment (BRWRE) on Cayley Graphs in recurrence and transience. This classification is stated explicitly for BRWRE on $\Z^d.$ Furthermore, we emphasize the interplay between Branching Markov Chains and the spectral radius. We prove properties of the spectral radius of the Random Walk in Random Environment with the help of appropriate Branching Markov Chains. http://front.math.ucdavis.edu/math.PR/0606055 --------------------------------------------------------------- 4347. THE KNEE-JERK MAPPING Peter G. Doyle and Jim Reeds We claim to give the definitive theory of what we call the `knee-jerk mapping', which is the basis for a class of optimization algorithms introduced by Baum, and promoted by Dempster, Laird, and Rubin under the name `EM algorithm'. http://front.math.ucdavis.edu/math.PR/0606068 --------------------------------------------------------------- 4348. WIENER INTEGRALS, MALLIAVIN CALCULUS AND COVARIANCE MEASURE STRUCTURE Ida Kruk (LAGA) and Francesco Russo (LAGA) and Ciprian Tudor (SAMOS) We introduce the notion of {\em covariance measure structure} for square integrable stochastic processes. We define Wiener integral, we develop a suitable formalism for stochastic calculus of variations and we make Gaussian assumptions only when necessary. Our main examples are finite quadratric variation processes with stationary increments and the bifractional Brownian motion. http://front.math.ucdavis.edu/math.PR/0606069 --------------------------------------------------------------- 4349. Q-GENERALIZATION OF SYMMETRIC ALPHA-STABLE DISTRIBUTIONS. PART I Sabir Umarov and Constantino Tsallis and Murray Gell-Mann and Stanly Steinberg The classic and the L\'evy-Gnedenko central limit theorems play a key role in theory of probabilities, and also in Boltzmann-Gibbs (BG) statistical mechanics. They both concern the paradigmatic case of probabilistic independence of the random variables that are being summed. A generalization of the BG theory, usually referred to as nonextensive statistical mechanics and characterized by the index $q$ ($q=1$ recovers the BG theory), introduces global correlations between the random variables, and recovers independence for $q=1$. The classic central limit theorem was recently $q$-generalized by some of us. In the present paper we $q$-generalize the L\'evy-Gnedenko central limit theorem. http://front.math.ucdavis.edu/cond-mat/0606038 --------------------------------------------------------------- 4350. Q-GENERALIZATION OF SYMMETRIC ALPHA-STABLE DISTRIBUTIONS. PART II Sabir Umarov and Constantino Tsallis and Murray Gell-Mann and Stanly Steinberg The classic and the L\'evy-Gnedenko central limit theorems play a key role in theory of probabilities, and also in Boltzmann-Gibbs (BG) statistical mechanics. They both concern the paradigmatic case of probabilistic independence of the random variables that are being summed. A generalization of the BG theory, usually referred to as nonextensive statistical mechanics and characterized by the index $q$ ($q=1$ recovers the BG theory), introduces global correlations between the random variables, and recovers independence for $q=1$. The classic central limit theorem was recently $q$-generalized by some of us. In the present paper we $q$-generalize the L\'evy-Gnedenko central limit theorem. In Part I we described the $q$-version of the $\alpha$- stable L\'evy distributions. In Part II we study the $(q^{\ast},q,q_{\ast})- $triplet, for which the mapping $F_{q^{\ast}}: \, \mathcal{G}_{q} \rightarrow \mathcal{G}_{q_{\ast}}$ holds. This fact allows to study the corresponding attractors and to obtain a complete generalization of the $q$-central limit theorem for random variables with infinite $(2q-1)$-variance. http://front.math.ucdavis.edu/cond-mat/0606040 --------------------------------------------------------------- 4351. SOME PROPERTIES OF EXPONENTIAL INTEGRALS OF L\'EVY PROCESSES AND EXAMPLES Hitoshi Kondo and Makoto Maejima and Ken-iti Sato The improper stochastic integral $Z=\int_0^{\infty-}\exp(-X_{s-})dY_s $ is studied, where $\{(X_t, Y_t), t \geqslant 0 \}$ is a L\'evy process on $\mathbb R ^{1+d}$ with $\{X_t \}$ and $\{Y_t \}$ being $\mathbb R$-valued and $\mathbb R ^d$-valued, respectively. The condition for existence and finiteness of $Z$ is given and then the law $\mathcal L(Z)$ of $Z$ is considered. Some sufficient conditions for $\mathcal L(Z)$ to be selfdecomposable and some sufficient conditions for $\mathcal L(Z)$ to be non-selfdecomposable but semi-selfdecomposable are given. Attention is paid to the case where $d=1$, $\{X_t\}$ is a Poisson process, and $\{X_t\}$ and $\{Y_t\}$ are independent. An example of $Z$ of type $G$ with selfdecomposable mixing distribution is given. http://front.math.ucdavis.edu/math.PR/0606084 --------------------------------------------------------------- 4352. HITTING TIMES FOR GAUSSIAN PROCESSES L. Decreusefond and D. Nualart We establish a general formula for the Laplace transform of the hitting times of a Gaussian process. Some consequences are derived, and in particular cases like the fractional Brownian motion are discussed. http://front.math.ucdavis.edu/math.PR/0606086 --------------------------------------------------------------- 4353. PROJECTION FORMULAS FOR ORTHOGONAL POLYNOMIALS W. Bryc and W. Matysiak and R. Szwarc and J. Wesolowski We prove a new projection formula for the four-parameter family of orthogonal polynomials outside of the Askey-Wilson class. By carefully analyzing the recurrence relations we manage to overcome the lack of explicit expression for the orthogonality measure. http://front.math.ucdavis.edu/math.CA/0606092 --------------------------------------------------------------- 4354. GAUSSIAN MARGINALS OF PROBABILITY MEASURES WITH GEOMETRIC SYMMETRIES Mark W. Meckes Motivated by the multivariate version of the central limit problem for convex bodies, we prove normal approximation theorems for k-dimensional marginals of probability measures on R^n possessing certain geometric symmetries. In particular, we derive results for uniform measures on 1-unconditional and 1-symmetric convex bodies and on simplices. We also discuss connections between results of E. Meckes and the author for 1-dimensional marginals and a recent result of B. Klartag. http://front.math.ucdavis.edu/math.MG/0606073 --------------------------------------------------------------- 4355. A DISCRETE INVITATION TO QUANTUM FILTERING AND FEEDBACK CONTROL Luc Bouten and Ramon van Handel and and Matthew R. James The engineering and control of devices at the quantum-mechanical level--such as those consisting of small numbers of atoms and photons--is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a novel field of application for stochastic estimation and control theory. In this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance. The model is extremely rich and allows a full development of the theory, while remaining completely within the setting of finite-dimensional Hilbert spaces (thus avoiding the technical complications of the continuous theory). We introduce discretized models of an atom in interaction with the electromagnetic field, obtain filtering equations for photon counting and homodyne detection, and solve a stochastic control problem using dynamic programming and Lyapunov function methods. http://front.math.ucdavis.edu/math.PR/0606118 --------------------------------------------------------------- 4356. PARAMETER-BASED FISHER'S INFORMATION OF ORTHOGONAL POLYNOMIALS J.S. Dehesa and B. Olmos & R.J. Yanez The Fisher information of the classical orthogonal polynomials with respect to a parameter is introduced, its interest justified and its explicit expression for the Jacobi, Laguerre, Gegenbauer and Grosjean polynomials found. http://front.math.ucdavis.edu/math.CA/0606133 --------------------------------------------------------------- 4357. DICHOTOMOUS MARKOV NOISE: EXACT RESULTS FOR OUT-OF-EQUILIBRIUM SYSTEMS (A BRIEF OVERVIEW) Ioana Bena Nonequilibrium systems driven by additive or multiplicative dichotomous Markov noise appear in a wide variety of physical and mathematical models. We review here some prototypical examples, with an emphasis on {\em analytically-solvable} situations. In particular, it has escaped attention till recently that the standard results for the long-time properties of such systems cannot be applied when unstable fixed points are crossed in the asymptotic regime. We show how calculations have to be modified to deal with these cases and present a few relevant applications -- the hypersensitive transport, the rocking ratchet, and the stochastic Stokes' drift. These results reinforce the impression that dichotomous noise can be put on a par with Gaussian white noise as far as obtaining analytic results is concerned. They convincingly illustrate the interplay between noise and nonlinearity in generating nontrivial behaviors of nonequilibrium systems and point to various practical applications. http://front.math.ucdavis.edu/cond-mat/0606116 --------------------------------------------------------------- 4358. PERCOLATION ON DUAL LATTICES WITH K-FOLD SYMMETRY Bela Bollobas and Oliver Riordan Zhang found a simple, elegant argument deducing the non-existence of an infinite open cluster in certain lattice percolation models (for example, p=1/2 bond percolation on the square lattice) from general results on the uniqueness of an infinite open cluster when it exists; this argument requires some symmetry. Here we show that a simple modification of Zhang's argument requires only 2-fold (or 3-fold) symmetry, proving that the critical probabilities for percolation on dual planar lattices with such symmetry sum to 1. We also give a new proof of a result of Grimmett determining the critical surface for anisotropic percolation on the triangular lattice. http://front.math.ucdavis.edu/math.PR/0606149 --------------------------------------------------------------- 4359. GENERALIZED CHEEGER INEQUALITIES FOR EIGENVALUES OF NON- REVERSIBLE MARKOV CHAINS Ravi Montenegro We show lower bounds for the smallest non-trivial eigenvalue, and smallest real portion of an eigenvalue, of the Laplacian of a non-reversible Markov chain in terms of an Evolving set quantity. A myriad of Cheeger-like inequalities follow for non-reversible chains, which even in the reversible case sharpen previously known results. The same argument also produces a new Cheeger-like inequality for the smallest eigenvalue of a reversible chain, and a Cheeger-like inequality for the second largest magnitude eigenvalue of a non-reversible chain. http://front.math.ucdavis.edu/math.PR/0606167 --------------------------------------------------------------- 4360. STUDENT'S T-TEST WITHOUT SYMMETRY CONDITIONS Iosif Pinelis An explicit representation of an arbitrary zero-mean distribution as the mixture of (at-most-)two-point zero-mean distributions is given. Based in this representation, tests for (i) asymmetry patterns and (ii) for location without symmetry conditions can be constructed. Exact inequalities implying conservative properties of such tests are presented. These developments extend results established earlier by Efron, Eaton, and Pinelis under a symmetry condition. http://front.math.ucdavis.edu/math.ST/0606160 --------------------------------------------------------------- 4361. CORRELATION DECAY AND DETERMINISTIC FPTAS FOR COUNTING LIST- COLORINGS OF A GRAPH David Gamarnik and Dmitriy Katz We propose a deterministic algorithm for approximately counting the number of list colorings of a graph. Under the assumption that the graph is triangle free, the size of every list is at least $\alpha \Delta$, where $ \alpha$ is an arbitrary constant bigger than $\alpha^{**}=2.8432...$, the solution of $\alpha e^{-{1\over \alpha}}=2$, and $\Delta$ is the maximum degree of the graph, we obtain the following results. For the case when the size of the each list is a large constant, we show the existence of a \emph{deterministic} FPTAS for computing the total number of list colorings. The same deterministic algorithm has complexity $2^{O(\log^2 n)}$, without any assumptions on the sizes of the lists, where $n$ is the size of the instance. Our results are not based on the most powerful existing counting technique -- rapidly mixing Markov chain method. Rather we build upon concepts from statistical physics, in particular, the decay of correlation phenomena and its implication for the uniqueness of Gibbs measures in infinite graphs. This approach was proposed in two recent papers \cite {BandyopadhyayGamarnikCounting} and \cite{weitzCounting}. The principle insight of the present work is that the correlation decay property can be established with respect to certain \emph{computation tree}, as opposed to the conventional correlation decay property which is typically established with respect to graph theoretic neighborhoods of a given node. This allows truncation of computation at a logarithmic depth in order to obtain polynomial accuracy in polynomial time. While the analysis conducted in this paper is limited to the problem of counting list colorings, the proposed algorithm can be extended to an arbitrary constraint satisfaction problem in a straightforward way. http://front.math.ucdavis.edu/math.CO/0606143 --------------------------------------------------------------- 4362. TRUELS, OR THE SURVIVAL OF THE WEAKEST Pau Amengual and Ra\'ul Toral In this paper we review some of the main results obtained in the field of truels. A "truel" is a generalization of a duel involving three players. Depending on the rules used for chosing the players, we may distinguish between the random, sequential and simultaneous truel. A paradoxical result appears in these games, as the player with the highest marksmanship does not necessarily possess the highest survival (or winning) probability. In this work we limit ourselves to the random and sequential truels in which players use their best possible strategy with no coalitions. Furthermore, we have modified the random truel and converted it into an opinion model. In this version each of the three players holds a different opinion on a given topic. We address next the question of who wins a "truel league". We will see that, despite the paradoxical result mentioned above, still the distribution of winners is peaked around the players with the higher marksmanship for the random and opinion versions. In the sequential truel, however, the paradoxical result remains partially since the distribution of winners is peaked around the intermediate players. If the rules of truels are extended from three to $N$ players, the paradoxical results shows up even more clearly since as $N$ increases it is more difficult for the player with the highest marksmanship to win the game. Finally, we consider the dynamics of the games in a spatial distribution in a given network of interactions. http://front.math.ucdavis.edu/math.PR/0606181 --------------------------------------------------------------- 4363. GENERALIZATIONS OF HO-LEE'S BINOMIAL INTEREST RATE MODEL I: FROM ONE- TO MULTI-FACTOR Jir\^o Akahori and Hiroki Aoki and and Yoshihiko Nagata In this paper a multi-factor generalization of Ho-Lee model is proposed. In sharp contrast to the classical Ho-Lee, this generalization allows for those movements other than parallel shifts, while it still is described by a recombining tree, and is stationary to be compatible with principal component analysis. Based on the model, generalizations of duration-based hedging are proposed. A continuous-time limit of the model is also discussed. http://front.math.ucdavis.edu/math.PR/0606183 --------------------------------------------------------------- 4364. STABLE SEMIGROUPS ON HOMOGENEOUS TREES AND HYPERBOLIC SPACES Andrzej Stos We prove the kernel estimates related to subordinated semigroups on homogeneous trees. We study the long time propagation problem. We exploit this to show exit time estimates for (large) balls. We use an abstract setting of metric measure spaces. This enables us to give these results for trees end hyperbolic spaces as well. Finally, we show some estimates for the Poisson kernel of a ball. http://front.math.ucdavis.edu/math.PR/0606185 --------------------------------------------------------------- 4365. IDENTIFICATION D'UN PROCESSUS AUTOR\'{E}GRESSIF GAUSSIEN STABLE PAR LA M\'{E}THODE DE MOYENNISATION LOGARITHMIQUE DANS LE CAS R\'{E}EL Faouzi Chaabane (EASMS) and Hamdi Fathallah (LM-Versailles) In the present work, we consider a stable one-dimensional gaussian autoregressive model in continous time. Using the limit theorems with logarithmic averaging obtained for continous local martingales, we construct then an estimator of the noise covariance $\sigma^{2}$ and an estimator of $\theta$ different of the one of the least squares estimator. By exploiting the weighting method we ameliorate the convergence rates of these new estimators. http://front.math.ucdavis.edu/math.PR/0606200 --------------------------------------------------------------- 4366. FLOW PROPERTIES OF DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION L. Decreusefond and D. Nualart We prove that solutions of stochastic differential equations driven by fractional Brownian motion for $H>1/2$ define flows of homeomorphisms on $\mathbb{R}^{d}$. http://front.math.ucdavis.edu/math.PR/0606214 --------------------------------------------------------------- 4367. FREE JACOBI PROCESS Nizar Demni (PMA) and the PMA Collaboration Using a matrix approach, we define the free Jacobi process as the limit of the complex Jacobi matrix process. The we derive a free SDE which is analogous to its classical counterpart. To proceed, we prove that fro suitable parameters the process remains injective if it is initially injective and then use the polar decomposition. In the stationnary case, this will be easily deduced from the explicit expression of the spectral measure. In the general setting we derive a recurrence formula for the moments. Moreover, a p. d. e. for the Cauchy transform of the law is given. http://front.math.ucdavis.edu/math.PR/0606218 --------------------------------------------------------------- 4368. SIGNIFICANT EDGES IN THE CASE OF A NON-STATIONARY GAUSSIAN NOISE Isabelle Abraham (DCRE) and Romain Abraham (MAPMO) and Agnes Desolneux (MAP5), Sebastien Li-Thiao-Te (CMLA) In this paper, we propose an edge detection technique based on some local smoothing of the image followed by a statistical hypothesis testing on the gradient. An edge point being defined as a zero-crossing of the Laplacian, it is said to be a significant edge point if the gradient at this point is larger than a threshold $s(\eps)$ defined by: if the image $I$ is pure noise, then $\P(\norm{\nabla I}\geq s(\eps) \bigm| \Delta I = 0) \leq\eps$. In other words, a significant edge is an edge which has a very low probability to be there because of noise. We will show that the threshold $s(\eps)$ can be explicitly computed in the case of a stationary Gaussian noise. In images we are interested in, which are obtained by tomographic reconstruction from a radiograph, this method fails since the Gaussian noise is not stationary anymore. But in this case again, we will be able to give the law of the gradient conditionally on the zero-crossing of the Laplacian, and thus compute the threshold $s(\eps)$. We will end this paper with some experiments and compare the results with the ones obtained with some other methods of edge detection. http://front.math.ucdavis.edu/math.ST/0606219 --------------------------------------------------------------- 4369. A DISCRETE IT\^O CALCULUS APPROACH TO HE'S FRAMEWORK FOR MULTI- FACTOR DISCRETE MARKETS Jir\^o Akahori In the present paper, a discrete version of It\^o's formula for a class of multi-dimensional random walk is introduced and applied to the study of a discrete-time complete market model which we call He's framework. The formula unifies continuous-time and discrete-time settings and by regarding the latter as the finite difference scheme of the former, the order of convergence is obtained. The result shows that He's framework cannot be of order 1 scheme except for the one dimensional case. http://front.math.ucdavis.edu/math.PR/0606292 --------------------------------------------------------------- 4370. ON THE FREE ENERGY OF A DIRECTED POLYMER IN A BROWNIAN ENVIRONMENT John Moriarty and Neil O'Connell We prove a formula conjectured in O'Connell and Yor (2001) for the free energy density of a directed polymer in a Brownian environment in 1+1 dimensions. http://front.math.ucdavis.edu/math.PR/0606296 --------------------------------------------------------------- 4371. DYNAMICAL MODELS FOR CIRCLE COVERING: BROWNIAN MOTION AND POISSON UPDATING Johan Jonasson and Jeffrey Steif We consider two dynamical variants of the classical problem of random interval coverings of the unit circle, the latter having been completely solved by L. Shepp. In the first model, the centers of the intervals perform independent Brownian motions and in the second model, the positions of the intervals are updated according to independent Poisson processes where an interval of length l is updated at rate l^{-alpha} where alpha is a parameter. For the model with Brownian motions, a special case of our results is that if the length of the nth interval is c/n, then there are times at which a fixed point is not covered if and only if c <2 and there are times at which the circle is not fully covered if and only if c <3. For the Poisson updating model, we obtain analogous results with c t)=\nu(A)$, where $T_0^X$ is the hitting time of 0 of $X$. http://front.math.ucdavis.edu/math.PR/0606392 --------------------------------------------------------------- 4378. RATES OF CONVERGENCE OF A TRANSIENT DIFFUSION IN A SPECTRALLY NEGATIVE L\'{E}VY POTENTIAL Arvind Singh (PMA) We consider a diffusion process $X$ in a random L\'{e}vy potential $V $. We study the rates of convergence when the diffusion is transient under the assumption that the L\'{e}vy process does not possess positive jumps. We generalize the previous results of Hu-Shi-Yor (1999) for drifted Brownian potentials. In particular, we prove a conjecture of Carmona: provided that there exists $0<\kappa<1$ such that $E[e^{\kappa V\_1}]=1$, then $X\_t/t^\kappa$ converges to some non-degenerate distribution. These results are in a way analogous to those obtained by Kesten-Kozlov-Spitzer (1975) for the random walk in a random environment. http://front.math.ucdavis.edu/math.PR/0606411 --------------------------------------------------------------- 4379. THE RANK OF RANDOM GRAPHS Kevin P. Costello and Van H. Vu We show that almost surely the rank of the adjacency matrix of the Erd\"os-R\'enyi random graph $G(n,p)$ equals the number of non-isolated vertices for any $c\ln n/n