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 environ- ment. 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 con- sidering 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.