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 \partial_t-(\Delta/2-\nabla V^n \nabla) in a periodic potential V^n=\sum_{k=0}^n U_k(x/R_k) obtained as a superposition of Holder-continuous periodic potentials U_k (of period \T^d:=\R^d/\Z^d, d\in \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=d\omega_t -\nabla V^\infty(y_t) dt