Martingale Approach for Markov Processes in Random Environment and Branching Markov Chains

In this paper we study random walks with branching (BRW), and two examples of countable Markov chains in random environment which are a one dimensional random walk and a random string

Percolation for the stable marriage of Poisson and Lebesgue

Let $\Xi$ be the set of points (we call the elements of $\Xi$ centers) of Poisson process in $\R^d$, $d\geq 2$, with unit intensity. Consider the allocation of $\R^d$ to $\Xi$ which is stable in the sense of Gale-Shapley marriage problem and in which each center claims a region of volume $\alpha\leq 1$. We prove that there is no percolation in the set of claimed sites if $\alpha$ is small enough, and that, for high dimensions, there is percolation in the set of claimed sites if $\alpha<1$ is large enough.Comment: revised version (only minor correction since v2), 16 pages, 3 figure

Rate of escape and central limit theorem for the supercritical Lamperti problem

The study of discrete-time stochastic processes on the half-line with mean drift at x given by μ1(x)→0 as x→∞ is known as Lamperti’s problem. We give sharp almost-sure bounds for processes of this type in the case where μ1(x) is of order x−β for some β∈(0,1). The bounds are of order t1/(1+β), so the process is super-diffusive but sub-ballistic (has zero speed). We make minimal assumptions on the moments of the increments of the process (finiteness of (2+2β+ε)-moments for our main results, so fourth moments certainly suffice) and do not assume that the process is time-homogeneous or Markovian. In the case where xβμ1(x) has a finite positive limit, our results imply a strong law of large numbers, which strengthens and generalizes earlier results of Lamperti and Voit. We prove an accompanying central limit theorem, which appears to be new even in the case of a nearest-neighbour random walk, although our result is considerably more general. This answers a question of Lamperti. We also prove transience of the process under weaker conditions than those that we have previously seen in the literature. Most of our results also cover the case where β=0. We illustrate our results with applications to birth-and-death chains and to multi-dimensional non-homogeneous random walks

Logarithmic speeds for one-dimensional perturbed random walks in random environments

We study the random walk in a random environment on Z+={0,1,2,…}, where the environment is subject to a vanishing (random) perturbation. The two particular cases that we consider are: (i) a random walk in a random environment perturbed from Sinai’s regime; (ii) a simple random walk with a random perturbation. We give almost sure results on how far the random walker is from the origin, for almost every environment. We give both upper and lower almost sure bounds. These bounds are of order (logt)β, for β∈(1,∞), depending on the perturbation. In addition, in the ergodic cases, we give results on the rate of decay of the stationary distribution

Random walks in a quarter plane with zero drifts : transcience and recurrrence

Projet MEVALRésumé disponible dans le fichier PD

Classification of Markov chains describing the evolution of random strings

We consider LIFO queue with several customer types, batch arrival and batch services. We get necessary and sufficient conditions for this queue to be ergodic, null recurrent and transient. These are given in terms of positive solutions of finite system of equations pi = Fi(p1, ..., pk), i = 1, ..., k, where Fi are polynomials with positive coefficients

Random walks in a quarter plane with zero drifts. I : ergodicity and null recurrence

Projet MEVA

A local limit theorem for triple connections in subcritical Bernoulli percolation

We prove a local limit theorem for the probability of a site to be connected by disjoint paths to three points in subcritical Bernoulli percolation on $\mathbb{Z}^{d},\,d\geq2$ in the limit where their distances tend to infinity.Comment: 31 page

Random walks in two-dimensional complexes

Projet MEVA