Multiscale homogenization with bounded ratios and anomalous slow diffusion

Abstract

We show that the effective diffusivity matrix D(Vⁿ) for the heat operator ∂_t − (Δ/2 − ∇Vⁿ∇) in a periodic potential Vⁿ = ΣU_k(x/R_k) obtained as a superposition of Hölder-continuous periodic potentials U_k (of period ��^d := ℝ^d/ℤ^d, d ∈ ℕ^*, U_k(0) = 0) decays exponentially fast with the number of scales when the scale ratios R_k+1/R_k are bounded above and below. From this we deduce the anomalous slow behavior for a Brownian motion in a potential obtained as a superposition of an infinite number of scales, dy_t = dω_t − ∇V^∞(y_t)dt. © 2002 Wiley Periodicals, Inc.