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Probability Abstracts 90
This document contains abstracts 3796-3953 from
Nov-1-2005 to Dec-29-2005.
They have been mailed on January 4, 2006.
Author(s): David Anderson and Jonathan Mattingly and H. Frederik Nijhout and Michael Reed
Abstract: 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://arXiv.org/abs/math/0510642
http://front.math.ucdavis.edu/math.PR/0510642
(alternate)
Author(s): Fedor Nazarov and Mikhail Sodin and Alexander Volberg
Abstract: 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://arXiv.org/abs/math/0510654
http://front.math.ucdavis.edu/math.CV/0510654
(alternate)
Author(s): Emmanuel C\'{e}pa (MAPMO) and Dominique L\'{e}pingle (MAPMO)
Abstract: Brownian particles in electrostatic interaction may pairwise collide when the
interaction parameter is small. But multiple collisions are never possible.
http://arXiv.org/abs/math/0511445
http://front.math.ucdavis.edu/math.PR/0511445
(alternate)
Author(s): Larry Goldstein and Yosef Rinott
Abstract: 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://arXiv.org/abs/math/0511427
http://front.math.ucdavis.edu/math.ST/0511427
(alternate)
Author(s): Olexandr Ganyushkin and Volodymyr Mazorchuk
Abstract: 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://arXiv.org/abs/math/0511431
http://front.math.ucdavis.edu/math.CO/0511431
(alternate)
Author(s): E. Daems and A.B.J. Kuijlaars
Abstract: 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://arXiv.org/abs/math/0511470
http://front.math.ucdavis.edu/math.CA/0511470
(alternate)
Author(s): George Kordzakhia and Steven Lalley
Abstract: 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://arXiv.org/abs/math/0511504
http://front.math.ucdavis.edu/math.PR/0511504
(alternate)
Author(s): Larry Goldstein
Abstract: 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://arXiv.org/abs/math/0511510
http://front.math.ucdavis.edu/math.PR/0511510
(alternate)
Author(s): Boris Tsirelson
Abstract: 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://arXiv.org/abs/math/0511011
http://front.math.ucdavis.edu/math.PR/0511011
(alternate)
Author(s): Anna Karczewska
Abstract: 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://arXiv.org/abs/math/0511512
http://front.math.ucdavis.edu/math.PR/0511512
(alternate)
Author(s): Jean-Francois Le Gall
Abstract: 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://arXiv.org/abs/math/0511515
http://front.math.ucdavis.edu/math.PR/0511515
(alternate)
Author(s): Hiroyuki Matsumoto Marc Yor
Abstract: 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://arXiv.org/abs/math/0511517
http://front.math.ucdavis.edu/math.PR/0511517
(alternate)
Author(s): Hiroyuki Matsumoto Marc Yor
Abstract: 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://arXiv.org/abs/math/0511519
http://front.math.ucdavis.edu/math.PR/0511519
(alternate)
Author(s): Peter Friz and Nicolas Victoir
Abstract: 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://arXiv.org/abs/math/0511520
http://front.math.ucdavis.edu/math.PR/0511520
(alternate)
Author(s): Gideon Amir and Ori Gurel-Gurevich and Eyal Lubetzky and Amit Singer
Abstract: 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://arXiv.org/abs/math/0511526
http://front.math.ucdavis.edu/math.PR/0511526
(alternate)
Author(s): Matyas Barczy and Gyula Pap
Abstract: 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://arXiv.org/abs/math/0511016
http://front.math.ucdavis.edu/math.PR/0511016
(alternate)
Author(s): Vlada Limic and Anja Sturm
Abstract: 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://arXiv.org/abs/math/0511536
http://front.math.ucdavis.edu/math.PR/0511536
(alternate)
Author(s): D. Feyel and A.S. Ustunel and M. Zakai
Abstract: 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://arXiv.org/abs/math/0511545
http://front.math.ucdavis.edu/math.PR/0511545
(alternate)
Author(s): Francesco Caravenna
Abstract: 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://arXiv.org/abs/math/0511561
http://front.math.ucdavis.edu/math.PR/0511561
(alternate)
Author(s): Francesco Caravenna and Giambattista Giacomin
Abstract: 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://arXiv.org/abs/math/0511562
http://front.math.ucdavis.edu/math.PR/0511562
(alternate)
Author(s): R.Brouwer
Abstract: 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://arXiv.org/abs/math/0511021
http://front.math.ucdavis.edu/math.PR/0511021
(alternate)
Author(s): L. C. Chen and F. Y. Wu
Abstract: 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://arXiv.org/abs/cond-mat/0511296
http://front.math.ucdavis.edu/cond-mat/0511296
(alternate)
Author(s): Harold Widom
Abstract: 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://arXiv.org/abs/math/0511533
http://front.math.ucdavis.edu/math.CO/0511533
(alternate)
Author(s): Elizabeth Meckes
Abstract: 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://arXiv.org/abs/math/0509441
http://front.math.ucdavis.edu/math.PR/0509441
(alternate)
Author(s): Larry Goldstein and Aihua Xia
Abstract: 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://arXiv.org/abs/math/0509444
http://front.math.ucdavis.edu/math.PR/0509444
(alternate)
Author(s): Luigi Manca
Abstract: 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://arXiv.org/abs/math/0509446
http://front.math.ucdavis.edu/math.PR/0509446
(alternate)
Author(s): D. Marinucci
Abstract: 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://arXiv.org/abs/math/0509430
http://front.math.ucdavis.edu/math.PR/0509430
(alternate)
Author(s): Francis Comets and Jeremy Quastel and Alejandro F. Ramirez
Abstract: We consider an interacting particle system on the one dimensional lattice
$\bf Z$ modeling combustion. The process depends on two integer parameters
$2\le a
http://arXiv.org/abs/math/0511025
http://front.math.ucdavis.edu/math.PR/0511025
(alternate)
Author(s): Wendelin Werner
Abstract: We show that there exists (up to multiplicative constants) a unique and
natural measure on simple loops on Riemann surfaces, such that the measure is
conformally invariant and also invariant under restriction (i.e. the measure on
a Riemann surface S' that is contained in another Riemann surface S, is just
the measure on S restricted to those loops that stay in S'). We then study some
of its properties and consequences concerning outer boundaries of critical
percolation clusters and Brownian loops.
http://arXiv.org/abs/math/0511605
http://front.math.ucdavis.edu/math.PR/0511605
(alternate)
Author(s): Rapha\"el Rossignol
Abstract: Threshold phenomena are investigated under a general approach, following
Talagrand, Friedgut and Kalai. The general upper bound for the threshold width
of symmetric monotone properties is improved. This follows from a new lower
bound on the maximal influence of a variable on a Boolean function. The method
of proof is based upon a well known logarithmic Sobolev inequality on the
discrete cube. This new bound is shown to be asymptotically optimal.
http://arXiv.org/abs/math/0511607
http://front.math.ucdavis.edu/math.PR/0511607
(alternate)
Author(s): Alexander Barvinok
Abstract: Given m positive integers R=(r_i), n positive integers C=(c_j) such that sum
r_i = sum c_j =N, and mn non-negative weights W=(w_ij), we consider the total
weight T(R, C; W) of non-negative integer matrices (contingency tables)
D=(d_ij) with the row sums r_i, column sums c_j, and the weight of D equal to
the product w_ij^{d_ij}$. We present a randomized algorithm of a polynomial in
N complexity which approximates T(R,C; W) within a factor of (2 pi N)^{-1/2} (2
pi t)^{N/2t} e^{N/12t^2} where t=max{min r_i, min c_j}. In many cases, this
approximation provides an asymptotically accurate estimate of ln T(R, C; W).
The idea of the algorithm is to express T(R,C; W) as the expectation of the
permanent of an NxN random matrix with exponentially distributed entries and
approximate the expectation by the integral of an efficiently computable
log-concave function on R^{mn}.
http://arXiv.org/abs/math/0511596
http://front.math.ucdavis.edu/math.CO/0511596
(alternate)
Author(s): Mark B. Villarino
Abstract: We prove the explicit formula for the probability of a run of r successes in
n trials.
http://arXiv.org/abs/math/0511652
http://front.math.ucdavis.edu/math.PR/0511652
(alternate)
Author(s): Ivan Nourdin (LPMA)
Abstract: In this paper, we will focus - in dimension one - on the SDEs of the type
dX\_t=s(X\_t)dB\_t+b(X\_t)dt where B is a fractional Brownian motion. Our
principal motivation is to describe one of the simplest theory - from our point
of view - allowing to study this SDE, and this for any Hurst index H between 0
and 1. We will consider several definitions of solution and we will study, for
each one of them, in which condition one has existence and uniqueness. Finally,
we will examine the convergence or not of the canonical scheme associated to
our SDE, when the integral with respect to fBm is defined using the
Russo-Vallois symmetric integral.
http://arXiv.org/abs/math/0511027
http://front.math.ucdavis.edu/math.PR/0511027
(alternate)
Author(s): David J. Aldous (U.C. Berkeley)
Abstract: Consider routing traffic on the $N \times N$ torus, simultaneously between
all source-destination pairs, to minimize the cost $\sum_e c(e)f^2(e)$, where
$f(e)$ is the volume of flow across edge $e$ and the $c(e)$ form an i.i.d.
random environment. We prove existence of a rescaled $N \to \infty$ limit
constant for minimum cost, by comparison with an appropriate analogous problem
about minimum-cost flows across a $M \times M$ subsquare of the lattice.
http://arXiv.org/abs/math/0511694
http://front.math.ucdavis.edu/math.PR/0511694
(alternate)
Author(s): Romain Abraham (MAPMO) and Jean-Francois Delmas (CERMICS)
Abstract: We consider the height process of a Levy process with no negative jumps, and
its associated continuous tree representation. Using Levy snake tools developed
by Duquesne and Le Gall, with an underlying Poisson process, we construct a
fragmentation process, which in the stable case corresponds to the self-similar
fragmentation described by Miermont. For the general fragmentation process we
compute a family of dislocation measures as well as the law of the size of a
tagged fragment. We also give a special Markov property for the snake which is
interesting in itself.
http://arXiv.org/abs/math/0511702
http://front.math.ucdavis.edu/math.PR/0511702
(alternate)
Author(s): Michael R\"ockner and Zeev Sobol
Abstract: We develop a new method to uniquely solve a large class of heat equations, so
called Kolmogorov equations in infinitely many variables. The equations are
analyzed in spaces of sequentially weakly continuous functions weighted by
proper (Lyapunov type) functions. This way for the first time the solutions are
constructed everywhere without exceptional sets for equations with possibly
non-locally Lipschitz drifts. Apart from general analytic interest, the main
motivation is to apply this to uniquely solve martingale problems in the sense
of Stroock-Varadhan given by stochastic partial differential equations from
hydrodynamics, such as the stochastic Navier-Stokes equations. In this paper
this is done in the case of the stochastic generalized Burgers equation.
Uniqueness is shown in the sense of Markov flows.
http://arXiv.org/abs/math/0511708
http://front.math.ucdavis.edu/math.PR/0511708
(alternate)
Author(s): Tomasz Bojdecki and Luis G. Gorostiza and Anna Talarczyk
Abstract: We prove a functional limit theorem for the rescaled occupation time
fluctuations of a (d,\alpha,\beta)-branching particle system (particles moving
in R^d according to a symmetric \alpha-stable Levy process, branching law in
the domain of attraction of a (1+\beta)-stable law, 0<\beta<1, uniform Poisson
initial state) in the case of intermediate dimensions, \alpha/\beta < d <
\alpha(1+\beta)/\beta. The limit is a process of the form K\lambda \xi, where K
is a constant, \lambda is the Lebesgue measure on R^d, and \xi =(\xi_t)_{t\geq
0} is a (1+\beta)-stable process which has long range dependence. There are two
long range dependence regimes, one for all \beta>d/(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://arXiv.org/abs/math/0511739
http://front.math.ucdavis.edu/math.PR/0511739
(alternate)
Author(s): P. Pfaffelhuber and A. Wakolbinger
Abstract: 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://arXiv.org/abs/math/0511743
http://front.math.ucdavis.edu/math.PR/0511743
(alternate)
Author(s): Luiz Renato Fontes Charles M. Newman
Abstract: 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://arXiv.org/abs/math/0511029
http://front.math.ucdavis.edu/math.PR/0511029
(alternate)
Author(s): Tomasz Bojdecki and Luis G. Gorostiza and Anna Talarczyk
Abstract: 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://arXiv.org/abs/math/0511745
http://front.math.ucdavis.edu/math.PR/0511745
(alternate)
Author(s): Franz Merkl and Silke Rolles
Abstract: 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://arXiv.org/abs/math/0511750
http://front.math.ucdavis.edu/math.PR/0511750
(alternate)
Author(s): Fran\c{c}ois Bolley (UMPA-ENSL)
Abstract: 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://arXiv.org/abs/math/0511752
http://front.math.ucdavis.edu/math.PR/0511752
(alternate)
Author(s): Marius Junge and Javier Parcet and Quanhua Xu
Abstract: 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://arXiv.org/abs/math/0511732
http://front.math.ucdavis.edu/math.OA/0511732
(alternate)
Author(s): Rinaldo Schinazi and Jason Schweinsberg
Abstract: 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://arXiv.org/abs/math/0512009
http://front.math.ucdavis.edu/math.PR/0512009
(alternate)
Author(s): Michael Krivelevich and Asaf Nachmias
Abstract: 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://arXiv.org/abs/math/0512004
http://front.math.ucdavis.edu/math.CO/0512004
(alternate)
Author(s): Michael Krivelevich and Asaf Nachmias
Abstract: 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://arXiv.org/abs/math/0512010
http://front.math.ucdavis.edu/math.CO/0512010
(alternate)
Author(s): Richard P. Stanley
Abstract: 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://arXiv.org/abs/math/0512035
http://front.math.ucdavis.edu/math.CO/0512035
(alternate)
Author(s): Laurent Goergen
Abstract: 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://arXiv.org/abs/math/0512061
http://front.math.ucdavis.edu/math.PR/0512061
(alternate)
Author(s): Nicolas Champagnat (WIAS)
Abstract: 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://arXiv.org/abs/math/0512063
http://front.math.ucdavis.edu/math.PR/0512063
(alternate)
Author(s): Michael R\"ockner and Feng-Yu Wang
Abstract: 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://arXiv.org/abs/math/0512100
http://front.math.ucdavis.edu/math.PR/0512100
(alternate)
Author(s): Endre Cs\'{a}ki and Ant\'{o}nia F\"{o}ldes and P\'al R\'ev\'esz
Abstract: 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://arXiv.org/abs/math/0511049
http://front.math.ucdavis.edu/math.PR/0511049
(alternate)
Author(s): Florent Benaych-Georges (DMA)
Abstract: 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://arXiv.org/abs/math/0512080
http://front.math.ucdavis.edu/math.OA/0512080
(alternate)
Author(s): Florent Benaych-Georges (DMA)
Abstract: 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://arXiv.org/abs/math/0512081
http://front.math.ucdavis.edu/math.OA/0512081
(alternate)
Author(s): Vyacheslav M. Abramov
Abstract: 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://arXiv.org/abs/math/0512118
http://front.math.ucdavis.edu/math.PR/0512118
(alternate)
Author(s): K. Debicki and A. B. Dieker and T. Rolski
Abstract: 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://arXiv.org/abs/math/0512119
http://front.math.ucdavis.edu/math.PR/0512119
(alternate)
Author(s): Ph. Barbe (CNRS) and W.P. McCormick (UGA)
Abstract: 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://arXiv.org/abs/math/0512141
http://front.math.ucdavis.edu/math.PR/0512141
(alternate)
Author(s): Fabrice Blache (LMA-Clermont)
Abstract: 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://arXiv.org/abs/math/0512145
http://front.math.ucdavis.edu/math.PR/0512145
(alternate)
Author(s): Adam Massey and Steven J. Miller and John Sinsheimer
Abstract: 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://arXiv.org/abs/math/0512146
http://front.math.ucdavis.edu/math.PR/0512146
(alternate)
Author(s): Jean Jacod (IMJ)
Abstract: 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://arXiv.org/abs/math/0511052
http://front.math.ucdavis.edu/math.PR/0511052
(alternate)
Author(s): Floyd E. Brown and Anant P. Godbole
Abstract: 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://arXiv.org/abs/math/0512177
http://front.math.ucdavis.edu/math.PR/0512177
(alternate)
Author(s): R. Munasinghe and R. Rajesh and R. Tribe and O. Zaboronski
Abstract: 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://arXiv.org/abs/math/0512179
http://front.math.ucdavis.edu/math.PR/0512179
(alternate)
Author(s): Thomas Liggett and Jeffrey Steif and Balint Toth
Abstract: 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://arXiv.org/abs/math/0512191
http://front.math.ucdavis.edu/math.PR/0512191
(alternate)
Author(s): Romain Abraham (MAPMO) and Jean-Francois Delmas (CERMICS)
Abstract: 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://arXiv.org/abs/math/0512195
http://front.math.ucdavis.edu/math.PR/0512195
(alternate)
Author(s): Youri Davydov and Ilya Molchanov and Sergei Zuyev
Abstract: 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://arXiv.org/abs/math/0512196
http://front.math.ucdavis.edu/math.PR/0512196
(alternate)
Author(s): Hongjie Dong and N.V. Krylov
Abstract: 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://arXiv.org/abs/math/0512200
http://front.math.ucdavis.edu/math.PR/0512200
(alternate)
Author(s): Asaf Nachmias and Yuval Peres
Abstract: 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://arXiv.org/abs/math/0512201
http://front.math.ucdavis.edu/math.PR/0512201
(alternate)
Author(s): Roberto Oliveira
Abstract: 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://arXiv.org/abs/math/0510648
http://front.math.ucdavis.edu/math.PR/0510648
(alternate)
Author(s): Alexis Devulder (PMA)
Abstract: 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://arXiv.org/abs/math/0511053
http://front.math.ucdavis.edu/math.PR/0511053
(alternate)
Author(s): Peter Friz and Nicolas Victoir
Abstract: 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://arXiv.org/abs/math/0512213
http://front.math.ucdavis.edu/math.PR/0512213
(alternate)
Author(s): Tomasz Szarek
Abstract: 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://arXiv.org/abs/math/0512221
http://front.math.ucdavis.edu/math.PR/0512221
(alternate)
Author(s): Peter Major
Abstract: 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://arXiv.org/abs/math/0512238
http://front.math.ucdavis.edu/math.PR/0512238
(alternate)
Author(s): R. G. Dolgoarshinnykh Steven P. Lalley
Abstract: 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://arXiv.org/abs/math/0512252
http://front.math.ucdavis.edu/math.PR/0512252
(alternate)
Author(s): Giuseppe Da Prato and Boris L. Rozovskii and Michael R\"ockner and Feng-Yu Wang
Abstract: 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://arXiv.org/abs/math/0512259
http://front.math.ucdavis.edu/math.PR/0512259
(alternate)
Author(s): Didier Piau
Abstract: 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://arXiv.org/abs/math/0511058
http://front.math.ucdavis.edu/math.PR/0511058
(alternate)
Author(s): Vladimir I. Bogachev and Michael R\"ockner and Stanislav V. Shaposhnikov
Abstract: 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://arXiv.org/abs/math/0512264
http://front.math.ucdavis.edu/math.PR/0512264
(alternate)
Author(s): V. P. Belavkin
Abstract: 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://arXiv.org/abs/math/0512265
http://front.math.ucdavis.edu/math.PR/0512265
(alternate)
Author(s): Viorel Barbu and Vladimir I. Bogachev and Giuseppe Da Prato and Michael R\"ockner
Abstract: 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://arXiv.org/abs/math/0512266
http://front.math.ucdavis.edu/math.PR/0512266
(alternate)
Author(s): S.F.Yashkov
Abstract: 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://arXiv.org/abs/math/0512281
http://front.math.ucdavis.edu/math.PR/0512281
(alternate)
Author(s): David H. Wolpert
Abstract: 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://arXiv.org/abs/nlin/0512015
http://front.math.ucdavis.edu/nlin.AO/0512015
(alternate)
Author(s): V. P. Belavkin
Abstract: 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://arXiv.org/abs/math/0512288
http://front.math.ucdavis.edu/math.PR/0512288
(alternate)
Author(s): V. P. Belavkin
Abstract: 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://arXiv.org/abs/math/0512289
http://front.math.ucdavis.edu/math.PR/0512289
(alternate)
Author(s): V. P. Belavkin
Abstract: 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://arXiv.org/abs/math/0512290
http://front.math.ucdavis.edu/math.PR/0512290
(alternate)
Author(s): T. Byczkowski and J. Malecki
Abstract: 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://arXiv.org/abs/math/0512294
http://front.math.ucdavis.edu/math.PR/0512294
(alternate)
Author(s): Gady Kozma
Abstract: 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://arXiv.org/abs/math/0511036
http://front.math.ucdavis.edu/math.CA/0511036
(alternate)
Author(s): Iosif Pinelis
Abstract: 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://arXiv.org/abs/math/0512301
http://front.math.ucdavis.edu/math.PR/0512301
(alternate)
Author(s): Maxim Krikun (IEC)
Abstract: 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://arXiv.org/abs/math/0512304
http://front.math.ucdavis.edu/math.PR/0512304
(alternate)
Author(s): Stefan Adams and Jean-Bernard Bru and Wolfgang Koenig
Abstract: 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://arXiv.org/abs/math/0512305
http://front.math.ucdavis.edu/math.PR/0512305
(alternate)
Author(s): Gopal K Basak and Amites Dasgupta
Abstract: 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://arXiv.org/abs/math/0512325
http://front.math.ucdavis.edu/math.PR/0512325
(alternate)
Author(s): Wilfrid Kendall
Abstract: 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://arXiv.org/abs/math/0512336
http://front.math.ucdavis.edu/math.PR/0512336
(alternate)
Author(s): V. P. Belavkin
Abstract: 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://arXiv.org/abs/math/0512360
http://front.math.ucdavis.edu/math.PR/0512360
(alternate)
Author(s): V. P. Belavkin
Abstract: 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://arXiv.org/abs/math/0512362
http://front.math.ucdavis.edu/math.PR/0512362
(alternate)
Author(s): Alexander I. Bufetov
Abstract: 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://arXiv.org/abs/math/0511035
http://front.math.ucdavis.edu/math.DS/0511035
(alternate)
Author(s): F den Hollander and S G Whittington
Abstract: In this paper we study a two-dimensional directed self-avoiding walk model of
a random copolymer in a random emulsion.
http://arXiv.org/abs/math/0512374
http://front.math.ucdavis.edu/math.PR/0512374
(alternate)
Author(s): Nathanael Berestycki (U.B.C.) and Jim Pitman (U.C. BERKELEY)
Abstract: 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://arXiv.org/abs/math/0512378
http://front.math.ucdavis.edu/math.PR/0512378
(alternate)
Author(s): Sebastian Mosbach and Amanda G. Turner
Abstract: 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://arXiv.org/abs/math/0512364
http://front.math.ucdavis.edu/math.NA/0512364
(alternate)
Author(s): Iosif Pinelis
Abstract: 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://arXiv.org/abs/math/0512382
http://front.math.ucdavis.edu/math.PR/0512382
(alternate)
Author(s): Jean-Rene Chazottes and Cristian Giardina and Frank Redig
Abstract: 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://arXiv.org/abs/math/0512386
http://front.math.ucdavis.edu/math.PR/0512386
(alternate)
Author(s): Fran\c{c}ois Simenhaus (PMA)
Abstract: 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://arXiv.org/abs/math/0512388
http://front.math.ucdavis.edu/math.PR/0512388
(alternate)
Author(s): Jean Mairesse and Fr\'ed\'eric Math\'eus
Abstract: 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://arXiv.org/abs/math/0512391
http://front.math.ucdavis.edu/math.PR/0512391
(alternate)
Author(s): M. Gregoratti
Abstract: 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://arXiv.org/abs/math/0512393
http://front.math.ucdavis.edu/math.PR/0512393
(alternate)
Author(s): L. Bertini and A. De Sole and D. Gabrielli and G. Jona-Lasinio and C. Landim
Abstract: 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://arXiv.org/abs/math/0512394
http://front.math.ucdavis.edu/math.PR/0512394
(alternate)
Author(s): B. de Tili\`ere
Abstract: 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://arXiv.org/abs/math/0512395
http://front.math.ucdavis.edu/math.PR/0512395
(alternate)
Author(s): Fabrice Blache (IAM)
Abstract: 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://arXiv.org/abs/math/0512403
http://front.math.ucdavis.edu/math.PR/0512403
(alternate)
Author(s): J. Martin Lindsay and Stephen J. Wills
Abstract: 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://arXiv.org/abs/math/0512398
http://front.math.ucdavis.edu/math.FA/0512398
(alternate)
Author(s): A.M.Vershik and N.V.Tsilevich
Abstract: 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://arXiv.org/abs/math/0512389
http://front.math.ucdavis.edu/math.RT/0512389
(alternate)
Author(s): V. P. Belavkin
Abstract: 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://arXiv.org/abs/math/0512410
http://front.math.ucdavis.edu/math.PR/0512410
(alternate)
Author(s): V. P. Belavkin
Abstract: 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://arXiv.org/abs/math/0512413
http://front.math.ucdavis.edu/math.PR/0512413
(alternate)
Author(s): Piotr Milos
Abstract: 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 a
http://arXiv.org/abs/math/0512414
http://front.math.ucdavis.edu/math.PR/0512414
(alternate)
Author(s): V. P. Belavkin
Abstract: A history and drama of the development of quantum probability theory is
outlined starting from the discovery of the Plank's constant exactly a 100
years ago. It is shown that before the rise of quantum mechanics 75 years ago,
the quantum theory had appeared first in the form of the statistics of quantum
thermal noise and quantum spontaneous jumps which have never been explained by
quantum mechanics. Moreover, the only reasonable probabilistic interpretation
of quantum theory put forward by Max Born was in fact in irreconcilable
contradiction with traditional mechanical reality and classical probabilistic
causality. This led to numerous quantum paradoxes, some of them due to the
great inventors of quantum theory such as Einstein and Schroedinger. They are
reconsidered in this paper from the modern quantum probabilistic point of view.
http://arXiv.org/abs/math/0512415
http://front.math.ucdavis.edu/math.PR/0512415
(alternate)
Author(s): Andrei Khrennikov
Abstract: We develop an analogue of probability theory for probabilities taking values
in topological groups. We generalize Kolmogorov's method of axiomatization of
probability theory: main distinguishing features of frequency probabilities are
taken as axioms in the measure-theoretic approach. We also present a review of
non-Kolmogorovian probabilistic models including models with negative, complex,
and $p$-adic valued probabilities. The latter model is discussed in details.
The introduction of $p$-adic (as well as more general non-Archimedean)
probabilities is one of the main motivations for consideration of generalized
probabilities taking values in topological groups which are distinct from the
field of real numbers. We discuss applications of non-Kolmogorovian models in
physics and cognitive sciences. An important part of this paper is devoted to
statistical interpretation of probabilities taking values in topological groups
(and in particular in non-Archimedean fields).
http://arXiv.org/abs/math/0512427
http://front.math.ucdavis.edu/math.PR/0512427
(alternate)
Author(s): Paavo Salminen and Pierre Vallois
Abstract: The joint distribution of maximum increase and decrease for Brownian motion
up to an independent exponential time is computed. This is achieved by
decomposing the Brownian path at the hitting times of the infimum and the
supremum before the exponential time. It is seen that an important element in
our formula is the distribution of the maximum decrease for the three
dimensional Bessel process with drift started from 0 and stopped at the first
hitting of a given level. From the joint distribution of the maximum increase
and decrease it is possible to calculate the correlation coefficient between
these at a fixed time and this is seen to be -0.47936... .
http://arXiv.org/abs/math/0512440
http://front.math.ucdavis.edu/math.PR/0512440
(alternate)
Author(s): Peter Constantin and Gautam Iyer
Abstract: In this paper we derive a representation of the deterministic 3-dimensional
Navier-Stokes equations based on stochastic Lagrangian paths. The particle
trajectories obey SDEs driven by a uniform Wiener process; the inviscid Weber
formula for the Euler equations of ideal fluids is used to recover the velocity
field. This method admits a self-contained proof of local existence for the
nonlinear stochastic system, and can be extended to formulate stochastic
representations of related hydrodynamic-type equations, including viscous
Burgers equations and LANS-alpha models.
http://arXiv.org/abs/math/0511067
http://front.math.ucdavis.edu/math.PR/0511067
(alternate)
Author(s): Vivek S. Borkar
Abstract: This article gives an overview of the developments in controlled diffusion
processes, emphasizing key results regarding existence of optimal controls and
their characterization via dynamic programming for a variety of cost criteria
and structural assumptions. Stochastic maximum principle and control under
partial observations (equivalently, control of nonlinear filters) are also
discussed. Several other related topics are briefly sketched.
http://arXiv.org/abs/math/0511077
http://front.math.ucdavis.edu/math.PR/0511077
(alternate)
Author(s): Richard C. Bradley
Abstract: This is an update of, and a supplement to, a 1986 survey paper by the author
on basic properties of strong mixing conditions.
http://arXiv.org/abs/math/0511078
http://front.math.ucdavis.edu/math.PR/0511078
(alternate)
Author(s): S.A. Ladoucette and J.L. Teugels
Abstract: Let \{X_1, X_2, ...\} be a sequence of independent and identically
distributed positive random variables of Pareto-type with index \alpha>0 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://arXiv.org/abs/math/0511082
http://front.math.ucdavis.edu/math.PR/0511082
(alternate)
Author(s): Jacky Cresson (LM-Besan\c{c}on) and S\'{e}bastien Darses (LM-Besan\c{c}on)
Abstract: 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://arXiv.org/abs/math/0510655
http://front.math.ucdavis.edu/math.PR/0510655
(alternate)
Author(s): Giada Basile (CEREMADE) and Cedric Bernardin (UMPA-ENSL) and Stefano Olla (CEREMADE)
Abstract: 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://arXiv.org/abs/cond-mat/0509688
http://front.math.ucdavis.edu/cond-mat/0509688
(alternate)
Author(s): Ioannis Kontoyiannis (Athens U of Econ & Business) and Rami Zamir (Tel-Aviv University)
Abstract: 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://arXiv.org/abs/cs/0511009
http://front.math.ucdavis.edu/cs.IT/0511009
(alternate)
Author(s): Oliver Riordan
Abstract: 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://arXiv.org/abs/math/0511093
http://front.math.ucdavis.edu/math.CO/0511093
(alternate)
Author(s): Bernard Roynette (IEC) and Pierre Vallois (IEC) and Marc Yor (PMA)
Abstract: 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://arXiv.org/abs/math/0511102
http://front.math.ucdavis.edu/math.PR/0511102
(alternate)
Author(s): Frederik S Herzberg
Abstract: 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://arXiv.org/abs/math/0511106
http://front.math.ucdavis.edu/math.PR/0511106
(alternate)
Author(s): F. Klebaner and R. Liptser
Abstract: 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://arXiv.org/abs/math/0511116
http://front.math.ucdavis.edu/math.PR/0511116
(alternate)
Author(s): Jacky Cresson (LM-Besan\c{c}on) and S\'{e}bastien Darses (LM-Besan\c{c}on)
Abstract: 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://arXiv.org/abs/math/0510656
http://front.math.ucdavis.edu/math.PR/0510656
(alternate)
Author(s): Marc Malric (PMA)
Abstract: 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://arXiv.org/abs/math/0511154
http://front.math.ucdavis.edu/math.PR/0511154
(alternate)
Author(s): D. Brydges and R. van der Hofstad and W. Konig
Abstract: 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://arXiv.org/abs/math/0511169
http://front.math.ucdavis.edu/math.PR/0511169
(alternate)
Author(s): Mathilde Weill (DMA)
Abstract: 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://arXiv.org/abs/math/0511172
http://front.math.ucdavis.edu/math.PR/0511172
(alternate)
Author(s): Marcelo Ventura Freire and Serguei Popov and Marina Vachkovskaia
Abstract: 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://arXiv.org/abs/math/0511186
http://front.math.ucdavis.edu/math.PR/0511186
(alternate)
Author(s): S Sherly and M K Jose and E Sandhya and N Raju
Abstract: 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://arXiv.org/abs/math/0510658
http://front.math.ucdavis.edu/math.PR/0510658
(alternate)
Author(s): Magnus Bordewich and Martin Dyer and Marek Karpinski
Abstract: 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://arXiv.org/abs/math/0511202
http://front.math.ucdavis.edu/math.PR/0511202
(alternate)
Author(s): Antar Bandyopadhyay
Abstract: 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://arXiv.org/abs/math/0511203
http://front.math.ucdavis.edu/math.PR/0511203
(alternate)
Author(s): Harald Luschgy and Gilles Pag\`{e}s (PMA)
Abstract: 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://arXiv.org/abs/math/0511208
http://front.math.ucdavis.edu/math.PR/0511208
(alternate)
Author(s): Terence Tao and Van Vu
Abstract: 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://arXiv.org/abs/math/0511215
http://front.math.ucdavis.edu/math.PR/0511215
(alternate)
Author(s): Luc Bouten and Ramon van Handel
Abstract: 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://arXiv.org/abs/math-ph/0511021
http://front.math.ucdavis.edu/math-ph/0511021
(alternate)
Author(s): M. R. Grasselli
Abstract: 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://arXiv.org/abs/math/0511234
http://front.math.ucdavis.edu/math.ST/0511234
(alternate)
Author(s): Roberto Oliveira and Joel Spencer
Abstract: 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://arXiv.org/abs/math/0510663
http://front.math.ucdavis.edu/math.PR/0510663
(alternate)
Author(s): Daniel J. Ford
Abstract: 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://arXiv.org/abs/math/0511246
http://front.math.ucdavis.edu/math.PR/0511246
(alternate)
Author(s): M. D. Jara
Abstract: 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://arXiv.org/abs/math/0511249
http://front.math.ucdavis.edu/math.PR/0511249
(alternate)
Author(s): Patrick Cattiaux (MODAL'X and CMAP) and Ivan Gentil (CEREMADE) and Arnaud Guillin (CEREMADE)
Abstract: 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://arXiv.org/abs/math/0511255
http://front.math.ucdavis.edu/math.PR/0511255
(alternate)
Author(s): Jean Bertoin and Marc Yor
Abstract: 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://arXiv.org/abs/math/0511265
http://front.math.ucdavis.edu/math.PR/0511265
(alternate)
Author(s): Edward C. Waymire
Abstract: 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://arXiv.org/abs/math/0511266
http://front.math.ucdavis.edu/math.PR/0511266
(alternate)
Author(s): Wendelin Werner
Abstract: 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://arXiv.org/abs/math/0511268
http://front.math.ucdavis.edu/math.PR/0511268
(alternate)
Author(s): Nasir Ganikhodjaev and Hasan Akin and Farrukh Mukhamedov
Abstract: 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://arXiv.org/abs/math/0511270
http://front.math.ucdavis.edu/math.PR/0511270
(alternate)
Author(s): Dan Crisan and Jie Xiong
Abstract: 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://arXiv.org/abs/math/0510668
http://front.math.ucdavis.edu/math.PR/0510668
(alternate)
Author(s): Randal Douc (CMAP) and Eric Moulines (LTCI) and Philippe Soulier (MODAL'X)
Abstract: 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://arXiv.org/abs/math/0511273
http://front.math.ucdavis.edu/math.PR/0511273
(alternate)
Author(s): Marko Znidaric
Abstract: 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://arXiv.org/abs/math/0511226
http://front.math.ucdavis.edu/math.ST/0511226
(alternate)
Author(s): M. Balazs and F. Rassoul-Agha and T. Seppalainen and S. Sethuraman
Abstract: 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://arXiv.org/abs/math/0511287
http://front.math.ucdavis.edu/math.PR/0511287
(alternate)
Author(s): Masayuki Kumon and Akimichi Takemura and Kei Takeuchi
Abstract: 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://arXiv.org/abs/math/0510662
http://front.math.ucdavis.edu/math.ST/0510662
(alternate)
Author(s): Alex D. Gottlieb
Abstract: 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://arXiv.org/abs/math/0511334
http://front.math.ucdavis.edu/math.PR/0511334
(alternate)
Author(s): Paavo Salminen and Marc Yor (PMA)
Abstract: 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://arXiv.org/abs/math/0511336
http://front.math.ucdavis.edu/math.PR/0511336
(alternate)
Author(s): Dimitri Petritis (IRMAR)
Abstract: 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://arXiv.org/abs/math/0511346
http://front.math.ucdavis.edu/math.PR/0511346
(alternate)
Author(s): Steve Tanner
Abstract: 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://arXiv.org/abs/math/0511368
http://front.math.ucdavis.edu/math.PR/0511368
(alternate)
Author(s): Francesco Caravenna and Giambattista Giacomin and Lorenzo Zambotti
Abstract: 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://arXiv.org/abs/math/0511376
http://front.math.ucdavis.edu/math.PR/0511376
(alternate)
Author(s): Akira Sakai
Abstract: 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://arXiv.org/abs/math-ph/0510093
http://front.math.ucdavis.edu/math-ph/0510093
(alternate)
Author(s): Boris L. Granovsky
Abstract: 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://arXiv.org/abs/math/0511381
http://front.math.ucdavis.edu/math.PR/0511381
(alternate)
Author(s): Ivan Nourdin (PMA) and Ciprian A. Tudor (SAMOS)
Abstract: 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://arXiv.org/abs/math/0511383
http://front.math.ucdavis.edu/math.PR/0511383
(alternate)
Author(s): Anne-Laure Basdevant (PMA)
Abstract: 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://arXiv.org/abs/math/0511388
http://front.math.ucdavis.edu/math.PR/0511388
(alternate)
Author(s): Nathanael Enriquez (PMA)
Abstract: 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://arXiv.org/abs/math/0511402
http://front.math.ucdavis.edu/math.PR/0511402
(alternate)
Author(s): Marius Junge and Javier Parcet
Abstract: 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://arXiv.org/abs/math/0511406
http://front.math.ucdavis.edu/math.OA/0511406
(alternate)
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