Long-Time tails in a random diffusion model

Letw = {w(x)xZ^d} be a positive random field with i.i.d. distribution. Given its realization, letX_t be the position at timet of a particle starting at the origin and performing a simple random walk with jump rate w^–1(X_t). The processX={X_t:t0} combined withw on a common probability space is an example of random walk in random environment. We consider the quantities_t=(d/dt) E(X_t² –M^–1t and_t(w) = (d/dt)E_w(X_t²– M^– 1t). Here E_w. is expectation overX at fixedw and E = E_w (dw) is the expectation over bothX andw. We prove the following long-time tail results: (1) lim_t t^d/2_t= V²M^d/2–3(d/2)^d/2 and (2) lim_t t^d/4_st(w)= Z_s weakly in path space, with {Z_s:s>0} the Gaussian process with EZ_s=0 and EZ_rZ_s= V²M^d/2–4(d)^d/2 (r + s)^–d/2. HereM and V² are the mean and variance of w(0) under . The main surprise is that fixingw changes the power of the long-time tail fromd/2 tod/4. Since, with₀ the stationary measure for the environment process, our result (1) exhibits a long-time tail in an equilibrium autocorrelation function.