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

Abstract

We consider a model of random walk on ℤν, ν≥2, in a dynamical random environment described by a field ξ={ξ t (x): (t,x)∈ℤν+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) ∈ℤν+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.