Related papers: Large deviations for maximum local time of simple …

We study the rate of convergence to equilibrium of the self-repellent random walk and its local time process on the discrete circle $\mathbb{Z}_n$. While the self-repellent random walk alone is non-Markovian since the jump rates depend on…

We study continuous-time (variable speed) random walks in random environments on $\mathbb{Z}^d$, $d\ge2$, where, at time $t$, the walk at $x$ jumps across edge $(x,y)$ at time-dependent rate $a_t(x,y)$. The rates, which we assume stationary…

We consider a symmetric random walk on the $\nu$-dimensional lattice, whose exit probability from the origin is modified by an antisymmetric perturbation and prove the local central limit theorem for this process. A short-range correction…

Let $(X_t, t \geq 0)$ be an $\alpha$-stable random walk with values in $\Z^d$. Let $l_t(x) = \int_0^t \delta_x(X_s) ds$ be its local time. For $p>1$, not necessarily integer, $I_t = \sum_x l_t^p(x)$ is the so-called $p$-fold self-…

We consider a one-dimensional diffusion process $X$ in a $(-\kappa/2)$-drifted Brownian potential for $\kappa\neq 0$. We are interested in the maximum of its local time, and study its almost sure asymptotic behaviour, which is proved to be…

We study the first passage times of discrete-time branching random walks in ${\mathbb R}^d$ where $d\geq 1$. Here, the genealogy of the particles follows a supercritical Galton-Watson process. We provide asymptotics of the first passage…

We consider the precise upper large deviations estimates for the maximal displacement of a branching random walk. In addition, we obtain a description of the extremal process of the branching random walk conditioned on this large deviations…

Let \alpha ([0,1]^p) denote the intersection local time of p independent d-dimensional Brownian motions running up to the time 1. Under the conditions p(d-2)<d and d\ge 2, we prove lim_{t\to\infty}t^{-1}\log P\bigl{\alpha([0,1]^p)\ge…

The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walks, time changed by a discrete quadratic variation process. One basis of this is a similar…

Let $(X_t,t\geq 0)$ be a random walk on $\mathbb{Z}^d$. Let $ l_t(x)= \int_0^t \delta_x(X_s)ds$ be the local time at site $x$ and $ I_t= \sum\limits_{x\in\mathbb{Z}^d} l_t(x)^p $ the p-fold self-intersection local time (SILT). Becker and…

In \cite{SzT}, D. Sz\'asz and A. Telcs have shown that for the diffusively scaled, simple symmetric random walk, weak convergence to the Brownian motion holds even in the case of local impurities if $d \ge 2$. The extension of their result…

We consider Random Walk in Random Scenery, denoted $X_n$, where the random walk is symmetric on $Z^d$, with $d>4$, and the random field is made up of i.i.d random variables with a stretched exponential tail decay, with exponent $\alpha$…

We study the maximal displacement of branching random walks in a class of time inhomogeneous environments. Specifically, binary branching random walks with Gaussian increments will be considered, where the variances of the increments change…

We study, in d-dimensions, the random walker with geometrically shrinking step sizes at each hop. We emphasize the integrated quantities such as expectation values, cumulants and moments rather than a direct study of the probability…

Let (Z_n)_{n\in\N_0} be a d-dimensional random walk in random scenery, i.e., Z_n=\sum_{k=0}^{n-1}Y_{S_k} with (S_k)_{k\in\N_0} a random walk in Z^d and (Y_z)_{z\in Z^d} an i.i.d. scenery, independent of the walk. We assume that the random…

We study a random walk in a random environment (RWRE) on $\Z^d$, $1 \leq d < +\infty$. The main assumptions are that conditionned on the environment the random walk is reversible. Moreover we construct our environment in such a way that the…

In this article, we study a branching random walk in an environment which depends on the time. This time-inhomogeneous environment consists of a sequence of macroscopic time intervals, in each of which the law of reproduction remains…

Consider $M_n$ the maximal position at generation $n$ of a supercritical branching random walk. A\"id\'ekon (2013) obtained and described the convergence in law, as time $n$ goes to infinity, of $M_n-m_n$, where $m_n$ is an explicit…

In this paper we study the discrete approximation to Brownian motion with varying dimension (BMVD in abbreviation) introduced in [4] by continuous time random walks on square lattices. The state space of BMVD contains a $2$-dimensional…

In this paper we consider the field of local times of a discrete-time Markov chain on a general state space, and obtain uniform (in time) upper bounds on the total variation distance between this field and the one of a sequence of $n$…