Long-Time tails in a random diffusion model

Abstract

Letw = {w(x)∶x∃Z^d} be a positive random field with i.i.d. distributionΜ. 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:t⩾0} combined withw on a common probability space is an example of random walk in random environment. We consider the quantitiesΔ _t =(d/dt) E _μ (X ² _t −M ⁻¹ t andδ _t(w) = (d/dt)E_w(X ²_t − M⁻ 1t). Here E_w. is expectation overX at fixedw and E_Μ = ∫ E_w Μ(dw) is the expectation over bothX andw. We prove the following long-time tail results: (1) lim_t→∞ t^d/2δ_t= V²M^d/2−3(d/2π)^d/2 and (2) lim_{t → ∞} t^d/4 δ _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(dπ)^d/2 (r + s)^−d/2. HereM and V² are the mean and variance of w(0) under Μ. The main surprise is that fixingw changes the power of the long-time tail fromd/2 tod/4. Since$\Delta _t = ME_{\mu _0 } ([w^{ - 1} (X_0 ) - M^{ - 1} ][w^{ - 1} (X_t ) - M^{ - 1} ])$, withΜ ₀ the stationary measure for the environment process, our result (1) exhibits a long-time tail in an equilibrium autocorrelation function.