Discrete-time random motion in a continuous random medium

We propose a discrete-time random walk on , d=1,2,..., as a variant of recent models of random walk on 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 L2 estimates. We consider for brevity only the case of odd dimension d, as even dimension requires somewhat different estimates.