Discrete-time random motion in a continuous random medium

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