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

We consider a random walk on $\mathbb Z$ 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 $\asymp t^{-1/2}\epsilon^{-\alpha_1 t}$ for $\alpha_1>0$. The analysis shows that, as the parameters of the model vary, a transition to a fall-off of the type $\asymp\epsilon^{-\bar\alpha t}$, for $\bar\alpha\in(0,\alpha_1)$, may occur.