ALMOST-SURE CENTRAL-LIMIT-THEOREM FOR A MARKOV MODEL OF RANDOM-WALK IN DYNAMICAL RANDOM ENVIRONMENT

We consider a model of random walk on Z(v), v greater than or equal to 2, in a dynamical random environment described by a field xi = {xi(t) (x):(t, x) is an element of Z(v+1)}. The random walk transition probab ilities are taken as P(Xl+1 = y\X-t = x, xi(t) = eta) = P-0(y - x) + c (y - x; eta(x)). We assume that the variables {xi(t)(x):(t, x) is an e lement of Z(v+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 av erage, We prove that the C.L.T. holds almost-surely, with the same par ameters as for P-0, for all v greater than or equal to 2. For v greate r than or equal to 3 there is a finite random (i.e., dependent on xi) correction to the average of X-t, and there is a corresponding random correction of order O(1/root t) to the C.L.T.. For v greater than or e qual to 5 there is a finite random correction to the covariance matrix of X-t and a corresponding correction of order O(1/t) to the C.L.T.. Proofs are based on some new L-p estimates for a class of functionals of the field.