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^d, d≥ 2, in a dynamical random environment described by a field which is i.i.d. in space and time. We prove that the C.L.T. holds almost-surely, with the same parameters as for the average random walk. For d>2 there is a fnite random correction to the average of Xt, and for m >4 there is a finite random correction to the covariance matrix of Xt. Proofs are based on L^p estimates