An almost sure invariance principle for random walks in a space-time random environment

Abstract

We consider a discrete time random walk in a space-time i.i.d. random environment. We use a martingale approach to show that the walk is diffusive in almost every fixed environment. We improve on existing results by proving an invariance principle and considering environments with an L 2 averaged drift. We also state an a.s. invariance principle for random walks in general random environments whose hypothesis requires a subdiffusive bound on the variance of the quenched mean, under an ergodic invariant measure for the environment chain.

Invariance principle for the random conductance model

“…We will proceed as in [RS1] applying the theory of "fractional coboundaries" of Derriennic and Lin [DeLi]. Setting R n := χ(n, X n , ω) we need to show that…”

“…Another very successful approach is the well-established Kipnis-Varadhan technique based on the process of the environment as seen from the particle. In [RS1] this approach has been used to get a QFCLT for the random walk in space-time product environments. Moreover, it has been applied in [DoLi] to random walks in a dynamic enviroment, which forms a Gibbsian Markov chain in time with spatial mixing, and in [JR] to random walks on R d , where the environment is i.i.d.…”

“…Also in this paper we will follow the approach in [RS1], so we use the process of the environment as seen from the particle and the method of the 'corrector', that is we decompose the random walk X into a martingale and a time-dependent corrector function. Due to the time-inhomogeneity and the resulting lack of reversibility we need to apply the adaptions of the Kipnis-Varadhan method to non-reversible situations in [MW] and [KLO].…”

“…To that aim we still follow [RS1] and apply the theory of 'fractional coboundaries' of Derriennic and Lin in [DeLi]. The main step in this approach is to establish a subdiffusive bound on the corrector (see Proposition 3.1 below), which is done in Section 3.…”

An almost sure invariance principle for additive functionals of Markov chains

“…This note is a by-product of the authors' recent work on random walks in a random environment (Rassoul-Agha and Seppäläinen, 2005, 2006, 2007, where this invariance principle proved useful.…”