Asymptotic Decay of Correlations for a Random Walk on the Lattice $\mathbf Z^d$ in Interaction with a Markov Field

We consider a discrete-time random walk on $\mathbf Z^d$, $d=1,2,\dots$ in a random environment with Markov evolution in time. We complete and extend to all dimension $d\ge1$ our earlier results on the time decay of the correlations of the “environment from the point of view of the random walk”.