Multi-scale homogenization with bounded ratios and Anomalous Slow Diffusion

Abstract

We show that the effective diffusivity matrix D(V^n) for the heat operator ∂_t − (Δ/2 − ∇V^n∇) in a periodic potential V^n = Σ^n_(k=0)U_k(x/R_k) obtained as a superposition of Hölder-continuous periodic potentials U_k (of period T^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^∞(yt)dt.