We give necessary and sufficient conditions for the stationary density of semimartingale reflected Brownian motion in a wedge to be written as a finite sum of terms of exponential product form. Relying on geometric ideas reminiscent of the reflection principle, we give an explicit formula for the density in such cases.

Let $X=(X_t)_{t\geq 0}$ be a martingale and $H=(H_t)_{t\geq 0}$ be a predictable process taking values in $[-1,1]$. Let $Y$ denote the stochastic integral of $H$ with respect to $X$. We show that $$ ||\sup_{t\geq 0}Y_t||_1 \leq \beta_0 ||\sup_{t\geq 0}|X_t|||_1,$$ where $\beta_0=2,0856\ldots$ is the best possible. Furthermore, if, in addition, $X$ is nonnegative, then $$ ||\sup_{t\geq 0}Y_t||_1 \leq \beta_0^+ ||\sup_{t\geq 0}X_t||_1,$$ where $\beta_0^+=\frac{14}{9}$ is the best possible.

Consider a random walk in an irreducible random environment on the positive integers. We prove that the annealed law of the random walk determines uniquely the law of the random environment. An application to linearly edge-reinforced random walk is given.

A Markov chain is considered whose states are orderings of an underlying fixed tree and whose transitions are local ``random-to-front'' reorderings, driven by a probability distribution on subsets of the leaves. The eigenvalues of the transition matrix are determined using Brown's theory of random walk on semigroups.

Some examples of translation invariant site percolation processes on the $Z^2$ lattice are constructed, the most far-reaching example being one that satisfies uniform finite energy (meaning that the probability that a site is open given the status of all others is bounded away from 0 and 1) and exhibits a.s. the coexistence of an infinite open cluster and an infinite closed cluster. Essentially the same example shows that coexistence is possible between an infinite open cluster and an infinite closed cluster that are both robust under i.i.d. thinning.

We give a new representation of fractional Brownian motion with Hurst parameter $H\leq\frac{1}{2}$ using stochastic partial differential equations. This representation allows us to use the Markov property and time reversal, tools which are not usually available for fractional Brownian motion. We then give simple proofs that fractional Brownian motion does not hit points in the critical dimension, and that it does not have double points in the critical dimension. These facts were already known, but our proofs are quite simple and use some ideas of Lévy.

An Erratum is available in ECP volume 17 paper number 8 (http://dx.doi.org/10.1214/ECP.v17-1774)

Benjamini, Burdzy and Chen (2007) introduced the notion of a shy coupling: a coupling of a Markov process such that, for suitable starting points, there is a positive chance of the two component processes of the coupling staying at least a given positive distance away from each other for all time. Among other results, they showed that no shy couplings could exist for reflected Brownian motions in $C^2$ bounded convex planar domains whose boundaries contain no line segments. Here we use potential-theoretic methods to extend this Benjamini et al.(2007) result (a) to all bounded convex domains (whether planar and smooth or not) whose boundaries contain no line segments, (b) to all bounded convex planar domains regardless of further conditions on the boundary.

Let $[\mathcal{P}]$ be the points of a Poisson process on $R^d$ and $F$ a probability distribution with support on the non-negative integers. Models are formulated for generating translation invariant random graphs with vertex set $[\mathcal{P}]$ and iid vertex degrees with distribution $F$, and the length of the edges is analyzed. The main result is that finite mean for the total edge length per vertex is possible if and only if $F$ has finite moment of order $(d+1)/d$.

In this article, we will explore why Karlin-McGregor method of using orthogonal polynomials in the study of Markov processes was so successful for one dimensional nearest neighbor processes, but failed beyond nearest neighbor transitions. We will proceed by suggesting and testing possible fixtures.

This paper proves that the scaling limit of nearest-neighbour senile reinforced random walk is Brownian Motion when the time T spent on the first edge has finite mean. We show that under suitable conditions, when T has heavy tails the scaling limit is the so-called fractional kinetics process, a random time-change of Brownian motion. The proof uses the standard tools of time-change and invariance principles for additive functionals of Markov chains.

We prove a moment identity on the Wiener space that extends the Skorohod isometry to arbitrary powers of the Skorohod integral on the Wiener space. As simple consequences of this identity we obtain sufficient conditions for the Gaussianity of the law of the Skorohod integral and a recurrence relation for the moments of second order Wiener integrals. We also recover and extend the sufficient conditions for the invariance of the Wiener measure under random rotations given in A. S. Üstünel and M. Zakai Prob. Th. Rel. Fields 103 (1995), 409-429.

We consider the limit distribution of the orders of the $k$ largest components in the Erdos-Rényi random graph inside the "critical window" for arbitrary $k$. We prove a local limit theorem for this joint distribution and derive an exact expression for the joint probability density function.

We show that there exist Lévy processes $(X_t,t \geq 0)$ and reals $y\gt0$ such that for small $t$, the probability $P(X_t\gt y)$ has an expansion involving fractional powers or more general functions of $t$. This constrasts with previous results giving polynomial expansions under additional assumptions.

A Brownian motion observed at equidistant sampling points renders a random walk with normally distributed increments. For the difference between the expected maximum of the Brownian mo- tion and its sampled version, an expansion is derived with coefficients in terms of the drift, the Riemann zeta function and the normal distribution function.

In 1981, J. Hawkes conjectured the exact form of the Hausdorff gauge function for the boundary of supercritical Galton-Watson trees under a certain assumption on the tail at infinity of the total mass of the branching measure. Hawkes's conjecture has been proved by T. Watanabe in 2007 as well as other precise results on fractal properties of the boundary of Galton-Watson trees. The goal of this paper is to provide an elementary proof of Hawkes's conjecture under a less restrictive assumption than in T. Watanabe's paper, by use of size-biased Galton-Watson trees introduced by Lyons, Pemantle and Peres in 1995.

In this note we give semi-explicit solutions for $r$-balayages of matrix-exponential-Lévy processes. To this end, we turn to an identity for the joint Laplace transform of the first entry time and the undershoot and a semi-explicit solution of the negative Wiener-Hopf factor. Our result is closely related to the works by Mordecki in [11], Asmussen, Avram and Pistorius in [3], Chen, Lee and Sheu in [7], and many others

There is a result of Diaconis and Freedman which says that, in a limiting sense, for large collections of high-dimensional data most one-dimensional projections of the data are approximately Gaussian. This paper gives quantitative versions of that result. For a set of $n$ deterministic vectors $\{x_i\}$ in $R^d$ with $n$ and $d$ fixed, let $\theta$ be a random point of the sphere and let $\mu_\theta$ denote the random measure which puts equal mass at the projections of each of the $x_i$ onto the direction $\theta$. For a fixed bounded Lipschitz test function $f$, an explicit bound is derived for the probability that the integrals of $f$ with respect to $\mu_\theta$ and with respect to a suitable Gaussian distribution differ by more than $\epsilon$. A bound is also given for the probability that the bounded-Lipschitz distance between these two measures differs by more than $\epsilon$, which yields a lower bound on the waiting time to finding a non-Gaussian projection of the $x_i$, if directions are tried independently and uniformly.

This paper characterizes the density functions of absolutely continuous positive random variables with finite expectation whose respective distribution functions satisfy the so-called length-bias scaling property.

In this note we discuss the extension of the elementary stochastic Ito-integral w.r.t. an optional semimartingale. The paths of an optional semimartingale possess limits from the left and from the right, but may have double jumps. This leads to quite interesting phenomena in integration theory. <br> We find a mathematically tractable domain of general integrands. The simple integrands are embedded into this domain. Then, we characterize the integral as the unique continuous and linear extension of the elementary integral and show completeness of the space of integrals. Thus our integral possesses desirable properties to model dynamic trading gains in mathematical finance when security price processes follow optional semimartingales.

Several Itô formulas have been already established for Lévy processes. We explain according to which criteria they are not optimal and establish an extended Itô formula that satisfies that criteria. The interest, in particular, of this formula is to obtain the explicit decomposition of $F(X)$, for $X$ Lévy process and $F$ deterministic function with locally bounded first order Radon-Nikodym derivatives, as the sum of a Dirichlet process and a bounded variation process.

Some deviation inequalities and moderate deviation principles for the maximum likelihood estimators of parameters in an Ornstein-Uhlenbeck process with linear drift are established by the logarithmic Sobolev inequality and the exponential martingale method.

In this paper, we prove an almost sure limit theorem for the maxima of strongly dependent Gaussian sequences under some mild conditions. The result is an expansion of the weakly dependent result of E. Csaki and K. Gonchigdanzan.

We consider the first Dirichlet eigenvalue of diffusion operators on the half line. A criterion for the equivalence of the first Dirichlet eigenvalue with respect to the maximum domain and that to the minimum domain is presented. We also describle the relationships between the first Dirichlet eigenvalue of transient diffusion operators and the standard Muckenhoupt's conditions for the dual weighted Hardy inequality. Pinsky's result [17] and Chen's variational formulas [8] are reviewed, and both provide the original motivation for this research.

We prove the CLT for a random walk in a dynamical environment where the states of the environment at different sites are independent Markov chains.

In this article, we establish a probabilistic representation for the second-order moment of the solution of stochastic heat equation, with multiplicative noise, which is fractional in time and colored in space. This representation is similar to the one given in Dalang, Mueller and Tribe (2008) in the case of an s.p.d.e. driven by a Gaussian noise, which is white in time. Unlike the formula of Dalang, Mueller and Tribe (2008) ,which is based on the usual Poisson process, our representation is based on the planar Poisson process, due to the fractional component of the noise.

The edges of the complete bipartite graph $K_{n,n}$ are given independent exponentially distributed costs. Let $C_n$ be the minimum total cost of a perfect matching. It was conjectured by M. Mézard and G. Parisi in 1985, and proved by D. Aldous in 2000, that $C_n$ converges in probability to $\pi^2/6$. We give a short proof of this fact, consisting of a proof of the exact formula $1 + 1/4 + 1/9 + \dots + 1/n^2$ for the expectation of $C_n$, and a $O(1/n)$ bound on the variance.

Under iterated barycentric subdivision of a triangle, most triangles become flat in the sense that the largest angle tends to $\pi$. By analyzing a random walk on $SL_2(\mathbb{R})$ we give asymptotics with explicit constants for the number of flat triangles and the degree of flatness at a given stage of subdivision. In particular, we prove analytical bounds for the upper Lyapunov constant of the walk.

Using recent results on the behavior of multiple Wiener-Itô integrals based on Stein's method, we prove Hsu-Robbins and Spitzer's theorems for sequences of correlated random variables related to the increments of the fractional Brownian motion.

Models of random graphs are considered where the presence or absence of an edge depends on the random types (colours) of its vertices, so that whether or not edges are present can be dependent. The principal objective is to study large deviations in the number of edges. These graphs provide a natural example with two different non-degenerate large deviation regimes, one arising from large deviations in the colourings followed by typical edge placement and the other from large deviation in edge placement. A secondary objective is to illustrate the use of a general result on large deviations for mixtures.

The aim of this note is to prove estimates on mean values of the number of times that Itô processes observed at discrete times visit small balls in $\mathbb{R}^d$. Our technique, in the innite horizon case, is inspired by Krylov's arguments in [2, Chap.2]. In the finite horizon case, motivated by an application in stochastic numerics, we discount the number of visits by a locally exploding coeffcient, and our proof involves accurate properties of last passage times at 0 of one dimensional semimartingales.

We prove a strong form of the invariance under re-rooting of the distribution of the continuous random trees called Lévy trees. This expends previous results due to several authors.

Let us consider the maximum $M(D)$ of a Dyck path $D$ chosen uniformly in the set of Dyck paths with $2n$ steps. We prove that the exponential moment of $M(D)$ normalized by the square root of $n$ is bounded in the limit of infinite $n$. This uniform bound justifies an assumption used in literature to prove certain estimates of high moments of large random matrices.

We derive concentration inequalities for the spectral measure of large random matrices, allowing for certain forms of dependence. Our main focus is on empirical covariance (Wishart) matrices, but general symmetric random matrices are also considered.

It is well-known that a random variable, i.e. a function defined on a probability space, with values in a Borel space, can be represented on the special probability space consisting of the unit interval with Lebesgue measure. We show an extension of this to multivariate functions. This is motivated by some recent constructions of random graphs.

The mixer chain on a graph $G$ is the following Markov chain. Place tiles on the vertices of $G$, each tile labeled by its corresponding vertex. A "mixer" moves randomly on the graph, at each step either moving to a randomly chosen neighbor, or swapping the tile at its current position with some randomly chosen adjacent tile. We study the mixer chain on $\mathbb{Z}$, and show that at time $t$ the expected distance to the origin is $t^{3/4}$, up to constants. This is a new example of a random walk on a group with rate of escape strictly between $t^{1/2}$ and $t$.

This paper introduces and studies a family of new classes of infinitely divisible distributions on $\mathbb{R}^d$ with two parameters. Depending on parameters, these classes connect the Goldie-Steutel-Bondesson class and the class of generalized type $G$ distributions, connect the Thorin class and the class $M$, connect the class $M$ and the class of generalized type $G$ distributions. These classes are characterized by stochastic integral representations with respect to Lévy processes.

We prove the following exponential inequality for a pair of random variables $(A,B)$ with $B >0$ satisfying the canonical assumption, $E[\exp(\lambda A - \frac{\lambda^2}{2} B^2)]\leq 1$ for $\lambda \in R$, $$P\left( \frac{|A|}{\sqrt{ \frac{2q-1}{q} \left(B^2+ (E[|A|^p])^{2/p} \right) }} \geq x \right) \leq \left(\frac{q}{2q-1} \right)^{\frac{q}{2q-1}} x^{-\frac{q}{2q-1}} e^{-x^2/2} $$ for $x>0$, where $1/p+ 1/q =1$ and $p\geq1$. Applying this inequality, we obtain exponential bounds for the tail probabilities for self-normalized martingale difference sequences. We propose a method of hypothesis testing for the $L^p$-norm $(p \geq 1)$ of $A$ (in particular, martingales) and some stopping times. We apply this inequality to the stochastic TSP in $[0,1]^d$ ($d\geq 2$), connected to the CLT.

We prove that the density fluctuations for a zero-range process evolving on the $d$-dimensional supercritical percolation cluster, with $d\geq{3}$, are given by a generalized Ornstein-Uhlenbeck process in the space of distributions $\mathscr{S}'(\mathbb{R}^d)$.

We give a probabilistic interpretation for the Barnes $G$-function which appears in random matrix theory and in analytic number theory in the important moments conjecture due to Keating-Snaith for the Riemann zeta function, via the analogy with the characteristic polynomial of random unitary matrices. We show that the Mellin transform of the characteristic polynomial of random unitary matrices and the Barnes $G$-function are intimately related with products and sums of gamma, beta and log-gamma variables. In particular, we show that the law of the modulus of the characteristic polynomial of random unitary matrices can be expressed with the help of products of gamma or beta variables. This leads us to prove some non standard type of limit theorems for the logarithmic mean of the so called generalized gamma convolutions.

This paper considers the eigenvalues of symmetric Toeplitz matrices with independent random entries and band structure. We assume that the entries of the matrices have zero mean and a uniformly bounded 4th moment, and we study the limit of the eigenvalue distribution when both the size of the matrix and the width of the band with non-zero entries grow to infinity. It is shown that if the bandwidth/size ratio converges to zero, then the limit of the eigenvalue distributions is Gaussian. If the ratio converges to a positive limit, then the distributions converge to a non-Gaussian distribution, which depends only on the limit ratio. A formula for the fourth moment of this distribution is derived.

We consider the first-passage percolation problem on effectively one-dimensional graphs with vertex set $\{1,\dots,n\}\times\{0,1\}$ and translation-invariant edge-structure. For three of six non-trivial cases we obtain exact expressions for the asymptotic percolation rate $\chi$ by solving certain recursive distributional equations and invoking results from ergodic theory to identify $\chi$ as the expected asymptotic one-step growth of the first-passage time from $(0, 0)$ to $(n, 0)$.

The survival problem for a diffusing particle moving among random traps is considered. We introduce a simple argument to derive the quenched asymptotics of the survival probability from the Lifshitz tail effect for the associated operator. In particular, the upper bound is proved in fairly general settings and is shown to be sharp in the case of the Brownian motion among Poissonian obstacles. As an application, we derive the quenched asymptotics for the Brownian motion among traps distributed according to a random perturbation of the lattice.

We extend the Yamada-Watanabe condition for pathwise uniqueness to stochastic differential equations with jumps, in the special case where small jumps are summable.

We will formulate a type of Gauss' divergence formula on sets of functions which are greater than a specific value of which boundaries are not regular. Such formula was first established by L. Zambotti in 2002 with a profound study of stochastic processes. In this paper we will give a much shorter and simpler proof for his formula in a framework of the Malliavin calculus and give alternate expressions. Our approach also enables to show that such formulae hold in other Gaussian spaces.

In this paper we propose a refinement of Mineka's coupling inequality that gives a better upper bound for $d_{TV}\left({\cal L}\left(W\right),{\cal L}\left(W+1\right)\right)$, where $W$ is a sum of $n$ independent integer valued random variables, in the case when $\text{Var} W\gg n$.

For a coordinate symmetric random vector $(Y_1,\ldots,Y_n)={\bf Y} \in \mathbb{R}^n$, that is, one satisfying $(Y_1,\ldots,Y_n)=_d(e_1Y_1,\ldots,e_nY_n)$ for all $(e_1,\ldots,e_n) \in \{-1,1\}^n$, for which $P(Y_i=0)=0$ for all $i=1,2,\ldots,n$, the following Berry Esseen bound to the cumulative standard normal $\Phi$ for the standardized projection $W_\theta=Y_\theta/v_\theta$ of ${\bf Y}$ holds: $$ \sup_{x \in \mathbb{R}}|P(W_\theta \leq x) - \Phi(x)| \leq 2 \sum_{i=1}^n |\theta_i|^3 E| X_i|^3 + 8.4 E(V_\theta^2-1)^2, $$ where $Y_\theta=\theta \cdot {\bf Y}$ is the projection of ${\bf Y}$ in direction $\theta \in \mathbb{R}^n$ with $||\theta||=1$, $v_\theta=\sqrt{\mbox{Var}(Y_\theta)},X_i=|Y_i|/v_\theta$ and $V_\theta=\sum_{i=1}^n \theta_i^2 X_i^2$. As such coordinate symmetry arises in the study of projections of vectors chosen uniformly from the surface of convex bodies which have symmetries with respect to the coordinate planes, the main result is applied to a class of coordinate symmetric vectors which includes cone measure ${\cal C}_p^n$ on the $\ell_p^n$ sphere as a special case, resulting in a bound of order $\sum_{i=1}^n |\theta_i|^3$.

We prove an equality-in-law relating the maximum of GUE Dyson's Brownian motion and the non-colliding systems with a wall. This generalizes the well known relation between the maximum of a Brownian motion and a reflected Brownian motion

Let $M$ be an arbitrary Hermitian matrix of order $n$, and $k$ be a positive integer less than $n$. We show that if $k$ is large, the distribution of eigenvalues on the real line is almost the same for almost all principal submatrices of $M$ of order $k$. The proof uses results about random walks on symmetric groups and concentration of measure. In a similar way, we also show that almost all $k \times n$ submatrices of $M$ have almost the same distribution of singular values.

This paper provides a countable representation for a class of infinite-dimensional diffusions which extends the infinitely-many-neutral-alleles model and is related to the two-parameter Poisson-Dirichlet process. By means of Gibbs sampling procedures, we define a reversible Moran-type population process. The associated process of ranked relative frequencies of types is shown to converge in distribution to the two-parameter family of diffusions, which is stationary and ergodic with respect to the two-parameter Poisson-Dirichlet distribution. The construction provides interpretation for the limiting process in terms of individual dynamics.

We study the problem of directed polymers in gaussian environments in $\mathbb{Z}^d$ from the viewpoint of a gaussian family indexed by the set of random walk paths. In the zero-temperature case, we give a numerical bound on the maximum of the Hamiltonian, whereas in the finite temperature case, we establish an equivalence between the "very strong disorder" and the growth rate of the entropy associated to the model

The purpose of this note is to prove a central limit theorem for the $L^2$-modulus of continuity of the Brownian local time obtained in [3], using techniques of stochastic analysis. The main ingredients of the proof are an asymptotic version of Knight's theorem and the Clark-Ocone formula for the $L^2$-modulus of the Brownian local time

For an irreducible Markov chain $(X_n)_{n\ge 0}$ we identify the rate function governing the large deviation estimation of empirical mean $\frac {1}{n} \sum_{k=0}^{n-1} f(X_k)$ by means of the Donsker-Varadhan's entropy. That allows us to obtain the lower bound of large deviations for the empirical measure $\frac {1}{n} \sum_{k=0}^{n-1} \delta_{X_k}$ in full generality

We show that Varadhan's small time asymptotics for densities of the solution of a stochastic differential equation in $\mathbb{R}^n$ carries over to a Hilbert space-valued Ornstein-Uhlenbeck process whose transition semigroup is strongly Feller and symmetric. In the Hilbert space setting, densities are with respect to a Gaussian invariant measure.

We use a coupling method for functional stochastic differential equations with bounded memory to establish an analogue of Wang's dimension-free Harnack inequality [13]. The strong Feller property for the corresponding segment process is also obtained.

Schramm-Loewner Evolution describes the scaling limits of interfaces in certain statistical mechanical systems. These interfaces are geometric objects that are not equipped with a canonical parametrization. The standard parametrization of SLE is via half-plane capacity, which is a conformal measure of the size of a set in the reference upper half-plane. This has useful harmonic and complex analytic properties and makes SLE a time-homogeneous Markov process on conformal maps. In this note, we show that the half-plane capacity of a hull $A$ is comparable up to multiplicative constants to more geometric quantities, namely the area of the union of all balls centered in $A$ tangent to $R$, and the (Euclidean) area of a $1$-neighborhood of $A$ with respect to the hyperbolic metric.

We prove a moderate deviation principle for traces of words of weakly interacting random matrices defined by a multi-matrix model with a potential being a small perturbation of the GUE. The remarkable strength of high order expansions of the matrix model recently found by Guionnet and Maurel-Segala is the key fact that allows us to develop our result and provides also an alternative proof for a special case of the central limit theorem for traces of words, studied in the article of Guionnet and Maurel-Segala (2006).

We give a criterion for the non-reconstructability of tree-indexed $q$-state Markov chains obtained by broadcasting a signal from the root with a given transition matrix $M$. Non-reconstruction holds if $c(M)$ times the expected number of offspring on the Galton-Watson tree is smaller than 1. Here $c(M)$ is an explicit function, which is convex over the set of $M$'s with a given invariant distribution, that is defined in terms of a $(q-1)$-dimensional variational problem over symmetric entropies. This result is equivalent to proving the extremality of the free boundary condition Gibbs measure within the corresponding Gibbs-simplex. Our theorem holds for possibly non-reversible $M$ and its proof is based on a general recursion formula for expectations of a symmetrized relative entropy function, which invites their use as a Lyapunov function. In the case of the Potts model, the present theorem reproduces earlier results of the authors, with a simplified proof, in the case of the symmetric Ising model (where the argument becomes similar to the approach of Pemantle and Peres) the method produces the correct reconstruction threshold), in the case of the (strongly) asymmetric Ising model where the Kesten-Stigum bound is known to be not sharp the method provides improved numerical bounds.