LONG-TIME TAILS IN A RANDOM DIFFUSION-MODEL

Let w = {w(x): x is-an-element-of Z(d) be a positive random field with i.i.d. distribution mu. Given its realization, let X(t) be the position at time t of a particle starting at the origin and performing a simple random walk with jump rate w(-i) (X(t)). The process X = {X(t): t greater-than-or-equal-to 0} combined with w on a common probability space is an example of random walk in random environment. We consider the quantities DELTA(t) = (d/dt) E(mu)(X(t)2 - M-1t) and DELTA(t)(w) = (d/dt) E(w)(X(t)2 - M-1(t)). Here E(w) is expectation over X at fixed w and E(mu) = integral E(w)mu(dw) is the expectation over both X and w. We prove the following long-time tail results: (1) lim(t --> infinity) t(d/2) DELTA(t) = V2M(d/2-3)(d/2pi)d/2 and (2) lim(t --> infinity) t(d/4) DELTA(st)(w) = Z(s) weakly in path space, with {Z(s): s > 0} the Gaussian process with EZ(s) = 0 and EZ(r)Z(s) = V2M(d/2-4)(d/2pi)d/2 (r+s)-d/2. Here M and V2 are the mean and variance of w(0) under mu. The main surprise is that fixing w changes the power of the long-tine tail from d/2 to d/4. Since DELTA(t) = ME mu0 ([w-1(X0)-M-1][w-1(X(t)-M-1]), with mu0 the stationary measure for the environment process, our result (1) exhibits a long-time tail in an equilibrium autocorrelation function.