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.     

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.     

Diffusivity of a random walk on random walks

We consider a random walk with the constraint that each coordinate of the walk is at distance one from the following one. In this paper, we show that this random walk is slowed down by a variance factor with respect to the case of the classical simple random walk without constraint. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 267–283, 2015     

Random walk in a high density dynamic random environment

The goal of this note is to prove a law of large numbers for the empirical speed of a green particle that performs a random walk on top of a field of red particles which themselves perform independent simple random walks on Z^d${\mathbb{Z}}^d$, d≥1$d\ge 1$. The red particles jump at rate 1 and are in a Poisson equilibrium with density μ$\mu$. The green particle also jumps at rate 1, but uses different transition kernels p^′$p^{\prime }$ and p^″$p^{″}$ depending on whether it sees a red particle or not. It is shown that, in the limit as μ→∞$\mu \to \infty$, the speed of the green particle tends to the average jump under p^′$p^{\prime }$. This result is far from surprising, but it is non-trivial to prove. The proof that is given in this note is based on techniques that were developed in Kesten and Sidoravicius (2005) to deal with spread-of-infection models. The main difficulty is that, due to particle conservation, space–time correlations in the field of red particles decay slowly. This places the problem in a class of random walks in dynamic random environments for which scaling laws are hard to obtain.     

Large deviations for hitting times of a random walk in random environment on a strip

We consider a random walk in random environment on a strip, which is transient to the right. The random environment is stationary and ergodic. By the constructed enlarged random environment which was first introduced by Goldsheid （2008）, we obtain the large deviations conditioned on the environment （in the quenched case） for the hitting times of the random walk.     

The limit theorems for random walk with state space ℝ in a space-time random environment

We consider a discrete time random environment. We state that when the random walk on real number space in a environment is i.i.d., under the law, the law of large numbers, iterated law and CLT of the process are correct space-time random marginal annealed Using a martingale approach, we also state an a.s. invariance principle for random walks in general random environment whose hypothesis requires a subdiffusive bound on the variance of the quenched mean, under an ergodic invariant measure for the environment chain.     

Branching random walks with random environments in time

We consider a branching random walk on N with a random environment in time （denoted by ξ）. Let Zn be the counting measure of particles of generation n, and let Zn（t） be its Laplace transform. We show the convergence of the free energy n-llog Zn（t）, large deviation principles, and central limit theorems for the sequence of measures {Zn}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale Zn（t）/E[Zn（t）ξ].     

Asymptotic behaviors for percolation clusters with uncorrelated weights

We consider random processes occurring on bond percolation clusters and represented as a generalization of the “divide and color model” introduced by Häggström in 2001. We investigate the asymptotic behaviors for bond percolation clusters with uncorrelated weights. For subcritical and supercritical phases, we prove the law of large numbers and central limit theorems in the models corresponding to the so-called quenched and annealed probabilities.     

A mixed random walk on nonnegative matrices: A law of large numbers

In this paper a mixed random walk on nonnegative matrices has been studied. Under reasonable conditions, existence of a unique invariant probability measure and a law of large numbers have been established for such walks.     

Almost-sure central limit theorem for a Markov model of random walk in dynamical random environment

Summary We consider a model of random walk on ℤ^ν, ν≥2, in a dynamical random environment described by a field ξ={ξ _t (x): (t,x)∈ℤ^ν+1}. The random walk transition probabilities are taken as P(X _{t +1}= y|X _t = x,ξ _t =η) =P ₀( y−x)+ c(y−x;η(x)). We assume that the variables {ξ _t (x):(t,x) ∈ℤ^ν+1} are i.i.d., that both P ₀(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P ₀, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X _t , and there is a corresponding random correction of order to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X _t and a corresponding correction of order to the C.L.T.. Proofs are based on some new L ^p estimates for a class of functionals of the field. Received: 4 January 1996/In revised form: 26 May 1997     

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.     

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.     

Velocity and Lyapounov exponents of some random walks in random environment

We express the asymptotic velocity of random walks in random environment satisfying Kalikow's condition in terms of the Lyapounov exponents which have previously been used in the context of large deviations.     

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.     

Limit theorems for stationary random evolutions

On a separable Banach space, let A(ξ₁),A(ξ₂),... be a strictly stationary sequence of infinitesimal operators, centered so that EA(ξ_i) = 0, i = 1,2,.... This paper characterizes the limit of the random evolutions

as the solution to a martingale problem. This work is a direct extension of previous work on i.i.d. random evolutions.     

随机环境中带移民的上临界分枝过程的Berry-Esseen界

考虑独立同分布的随机环境中带移民的上临界分枝过程(Zn).应用(Zn)与随机环境中不带移民分枝过程的联系,以及与相应随机游动的联系,在一些适当的矩条件下,本文证明关于log Zn的中心极限定理的Berry-Esseen界.     

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.