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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)
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