Large deviations for a random walk in dynamical random environment

Abstract

We consider a random walk X t ϵ Z d , t ϵ Z +, in a dynamical random environment (ξ t ( x), x ϵ Z d ), t ϵ Z +, with a mutual interaction with each other. The Markov process ( X t , ξ t ( x), x ϵ Z d ) is a perturbation of a process for which the random walk X t and the environment ξ t ( x), x ϵ Z d are independent, X t , t ϵ Z + is a homogeneous random walk in Z d and the environment ξ t ( x), x ϵ Z d behaves independently in each site as an ergodic Markov chain. For the perturbated process we assume that 1. 1.|The interaction between the position of the particle and the environment is local; 2. 2.|The influence of the environment on the particle X t is small; 3. 3.|The particle modifies the environment of its location (it cancels the memory of the environment). We consider a large deviation problem for the random walk X t , t ϵ Z +. We prove that a large deviation principle holds for this random walk with a good rate function which is analytic with respect to the parameter of interaction in a neighborhood of 0.