Multi-scale homogenization with bounded ratios and Anomalous Slow Diffusion

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: $dy_t=d\omega_t -\nabla V^\infty(y_t) dt$