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 $\T^d:=\R^d/\Z^d$, $d\in {\mathbb{N}}^{\ast }$, $U_k(0)=0$) 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: $d{y}_t=d{\omega }_t-\nabla V^{\infty }(y_t)dt$

29 pages, 1 figure, submitted version