Related papers: Angular asymptotics for multi-dimensional non-homo…

The range, local times, and periodicity of symmetric, weakly asymmetric and asymmetric random walks at the time of exit from a strip with $N$ locations are considered. Several results on asymptotic distributions are obtained.

We first study a model, introduced recently in \cite{ES}, of a critical branching random walk in an IID random environment on the $d$-dimensional integer lattice. The walker performs critical (0-2) branching at a lattice point if and only…

We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails…

For a homogeneous random walk in the quarter plane with nearest-neighbor transitions, starting from some state $(i_0,j_0)$, we study the event that the walk reaches the vertical axis, before reaching the horizontal axis. We derive an exact…

Suppose we are given the free product V of a finite family of finite or countable sets. We consider a transient random walk on the free product arising naturally from a convex combination of random walks on the free factors. We prove the…

We consider random walks on discrete state spaces, such as general undirected graphs, where the random walkers are designed to approximate a target quantity over the network topology via sampling and neighborhood exploration in the form of…

In the first part of this thesis, we study a Markov chain on $\mathbb{R}_+ \times S$, where $\mathbb{R}_+$ is the non-negative real numbers and $S$ is a finite set, in which when the $\mathbb{R}_+$-coordinate is large, the $S$-coordinate of…

We derive asymptotic formulas for the mean exit time $\bar{\tau}^{N}$ of the fastest among $N$ identical independently distributed Brownian particles to an absorbing boundary for various initial distributions (partially uniformly and…

The entropy, the spectral radius and the drift are important numerical quantities associated to random walks on countable groups. We prove sharp inequalities relating those quantities for walks with a finite second moment, improving upon…

We analyze the differences between the horizontal and the vertical component of the simple random walk on the 2-dimensional comb. In particular we evaluate by combinatorial methods the asymptotic behaviour of the expected value of the…

We consider a one-dimensional Brownian motion of fixed duration $T$. Using a path-integral technique, we compute exactly the probability distribution of the difference $\tau=t_{\min}-t_{\max}$ between the time $t_{\min}$ of the global…

In this paper we identify the asymptotic tail of the distribution of the exit time $\tau_C$ from a cone $C$ of an isotropic $\alpha$-self-similar Markov process $X_t$ with a skew-product structure, that is $X_t$ is a product of its radial…

In this paper, we study random walks $g_n=f_{n-1}\cdots f_0$ on the group $\mathrm{Homeo}(S^1)$ of the homeomorphisms of the circle, where the homeomorphisms $f_k$ are chosen randomly, independently, with respect to a same probability…

We study variable-speed random walks on $\mathbb Z$ driven by a family of nearest-neighbor time-dependent random conductances $\{a_t(x,x+1)\colon x\in\mathbb Z, t\ge0\}$ whose law is assumed invariant and ergodic under space-time shifts. We…

Consider the random walk $S_n=\xi_1+...+\xi_n$ with independent and identically distributed increments and negative mean $\mathbf E\xi=-m<0$. Let $M=\sup_{0\le i} S_i$ be the supremum of the random walk. In this note we present derivation…

For a symmetric random walk in $Z^2$ which does not necessarily have bounded jumps we study those points which are visited an unusually large number of times. We prove the analogue of the Erd\H{o}s-Taylor conjecture and obtain the…

We investigate reflected random walks in the quarter plane, with particular emphasis on the time spent along the reflection boundary axes. Assuming the drift of the random walk lies within the cone, the local time converges -- without the…

We study a random walk (Markov chain) in an unbounded planar domain whose boundary is described by two curves of the form $x_2 = a^+ x_1^{\beta^+}$ and $x_2 = -a^- x_1^{\beta^-}$, with $x_1 \geq 0$. In the interior of the domain, the random…

By appealing to renewal theory we determine the equations that the mean exit time of a continuous-time random walk with drift satisfies both when the present coincides with a jump instant or when it does not. Particular attention is paid to…

We define the probability structure of a continuous-time time-homogeneous Markov jump process, on a finite graph, that represents the continuous-time counterpart of the so-called Ruelle-Bowen discrete-time random walk. It constitutes the…