一类随机环境下随机游动的常返性

就一类平稳环境θ下随机流动{Xn}n∈z 建立相应的Markov-双链{ηn}n∈z ={(xn,Tnθ)}n∈z ,并给出在该平稳环境θ下{xn}n∈z 为常返链的条件.     

Scaling limit theorem for transient random walk in random environment

We construct a sequence of transient random walks in random environments and prove that by proper scaling, it converges to a diffusion process with drifted Brownian potential. To this end, we prove a counterpart of convergence for transient random walk in non-random environment, which is interesting itself.     

Quenched moderate deviations principle for random walk in random environment

We derive a quenched moderate deviations principle for the one-dimensional nearest random walk in random environment, where the environment is assumed to be stationary and ergodic. The approach is based on hitting time decomposition.     

Tail estimates for one-dimensional non-nearest-neighbor random walk in random environment

Suppose that the integers are assigned i.i.d. random variables {(β gx , . . . , β 1x , α x )} (each taking values in the unit interval and the sum of them being 1), which serve as an environment. This environment defines a random walk {X n } (called RWRE) which, when at x, moves one step of length 1 to the right with probability α x and one step of length k to the left with probability β kx for 1≤ k≤ g. For certain environment distributions, we determine the almost-sure asymptotic speed of the RWRE and show that the chance of the RWRE deviating below this speed has a polynomial rate of decay. This is the generalization of the results by Dembo, Peres and Zeitouni in 1996. In the proof we use a large deviation result for the product of random matrices and some tail estimates and moment estimates for the total population size in a multi-type branching process with random environment.     

Shift-coupling and a zero-one law for random walk in random environment

This paper discusses several aspects of shift-coupling for random walk in random environment.     

Symmetric random walk in random environment in one dimension

We give a new proof of the central limit theorem for one dimensional symmetric random walk in random environment. The proof is quite elementary and natural. We show the convergence of the generators and from this we conclude the convergence of the process. We also investigate the hydrodynamic limit (HDL) of one dimensional symmetric simple exclusion in random environment and prove stochastic convergence of the scaled density field. The macroscopic behaviour of this field is given by a linear heat equation. The diffusion coefficient is the same as that of the corresponding random walk. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotic behavior for random walk in random environment with holding times

In this article, we mainly discuss the asymptotic behavior for multi-dimensional continuous-time random walk in random environment with holding times. By constructing a renewal structure and using the point “environment viewed from the particle”, under General Kalikow's Condition, we show the law of large numbers (LLN) and central limit theorem (CLT) for the escape speed of random walk.     

Limit theorem for a random walk in a reversible random environment

Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 29, No. 4, pp. 627–644, October–December, 1989.     

Slowdown and neutral pockets for a random walk in random environment

We consider a d-dimensional random walk in random environment for which transition probabilities at each site are either neutral or present an effective drift “pointing to the right”. We obtain large deviation estimates on the probability that the walk moves in a too slow ballistic fashion, both under the annealed and quenched measures. These estimates underline the key role of large neutral pockets of the medium in the occurrence of slowdowns of the walk. Received: 12 March 1998 / Revised version: 19 February 1999     

A monotonicity property for random walk in a partially random environment

We prove a law of large numbers for random walks in certain kinds of i.i.d. random environments in ${\mathbb{Z}}^d$ that is an extension of a result of Bolthausen et al. (2003) [4]. We use this result, along with the lace expansion for self-interacting random walks, to prove a monotonicity result for the first coordinate of the speed of the random walk under some strong assumptions on the distribution of the environment.     

Simple random walk on the line in random environment

Summary We obtain strong limiting bounds for the maximal excursion and for the maximum reached by a random walk in a random environment. Our results derive from a simple proof of Pólya's theorem for the recurrence of the random walk on the line. As applications, we obtain bounds for the number of visits of the random walk at the origin.     

An extension of Kesten's renewal theorem for random walk in a random environment

Kesten showed that for certain random walks in a random environment the distribution of the environment as seen from the vantage point of the random walker converges to a limit distribution for large time. It is shown here that under additional hypotheses Kesten's result continues to hold for almost every typical fixed environment.     

A remark on the CLT for a random walk in a random environment

We give a simplified proof, using elementary methods only, of the almost-sure central limit theorem (CLT) in any dimension for a Markov model of a random walk in a random environment introduced in [BMP].Mathematics Subject Classification (2000): 60F05, 60K37Revised version: 29 January 2004     

Precise large deviation estimates for a one-dimensional random walk in a random environment

ω_x } (taking values in the interval [1/2, 1)), which serve as an environment. This environment defines a random walk {X _k } (called a RWRE) which, when at x, moves one step to the right with probability ω _x , and one step to the left with probability 1 −ωx. Solomon (1975) determined the almost-sure asymptotic speed (= rate of escape) of a RWRE, in a more general set-up. Dembo, Peres and Zeitouni (1996), following earlier work by Greven and den Hollander (1994) on the quenched case, have computed rough tail asymptotics for the empirical mean of the annealed RWRE. They conjectured the form of the rate function in a full LDP. We prove in this paper their conjecture. The proof is based on a “coarse graining scheme” together with comparison techniques. Received: 22 July 1997/Revised version: 15 June 1998     

Central limit theorems for a branching random walk with a random environment in time

We consider a branching random walk with a random environment in time, in which the offspring distribution of a particle of generation n and the distribution of the displacements of its children depend on an environment indexed by the time n. The environment is supposed to be independent and identically distributed. For A ?, let Z_n(A) be the number of particles of generation n located in A. We show central limit theorems for the counting measure Z_n(·) with appropriate normalization.     

Limit law for transition probabilities and moderate deviations for Sinai's random walk in random environment

We consider a one-dimensional random walk in random environment in the Sinai's regime. Our main result is that logarithms of the transition probabilities, after a suitable rescaling, converge in distribution as time tends to infinity, to some functional of the Brownian motion. We compute the law of this functional when the initial and final points agree. Also, among other things, we estimate the probability of being at time t at distance at least z from the initial position, when z is larger than ln²t, but still of logarithmic order in time.Partially supported by CNRS (UMR 7599 ``Probabilités et Modéles Aléatoires'), and by the ``Réseau Mathématique France-Brésil'Partially supported by CNPq (300676/00–0 and 302981/02–0), COFECUB, and by the ``Rede Matemática Brasil-França'