Asymptotic decay of correlations for a random walk in interaction with a Markov field

We consider a random walk on Z in a random environment independent in space and with a Markov evolution in time. We study the decay in time of correlations of the increments of the annealed random walk. We prove that for small stochasticity they fall off as ˜t-1/2e-a1t, for a>0. The analysis shows that, as the parameters of the model vary, a transition to a fall-off of the type ˜e- \bar at, for \bar a ¿(0,a1), may occur.