Interacting random walk in a dynamical random environment. I. Decay of correlations

We consider a random walk on the d-dimensional lattice Z^d in mutual interaction with a random environment. The model is a small perturbation of an independent model in which random walk and environment are independent and the latter evolves at each site as an independent Markov chain. We consider the asymptotics in time of the correlations of the environment and prove that they decay as an inverse power