Discrete-time random motion in a continuous random medium

We propose a discrete-time random walk on RdRd, d=1,2,…d=1,2,…, as a variant of recent models of random walk on ZdZd in a random environment which is i.i.d. in space–time. We allow space correlations of the environment and develop an analytic method to deal with them. We prove, under some general assumptions, that if the random term is small, a “quenched” (i.e., for a fixed “history” of the environment) Central Limit Theorem for the displacement of the random walk holds almost-surely. Proofs are based on L2L2 estimates. We consider for brevity only the case of odd dimension dd, as even dimension requires somewhat different estimates.