Multi-scale homogenization with bounded ratios and Anomalous Slow Diffusion

This paper is concerned with the asymptotic behavior of effective diffusivity matrices $D(V_0^n)$, associated to media characterized by an arbitrarily large number of scales but with ratios bounded independently from their numbers, $V_0^n=\sum_{k=0}^n U_k(x/R_k)$, where $U_k$ are Holder-continuous functions of period $T^d_1$ (torus of dimension $d\geq 1$ and side 1), $U_k(0)=0$ and $R_k$ grows exponentially fast with $k$ but has bounded ratios $\sup_{k}R_{k+1}/R_k<\infty$. $$ ^tlD(V_0^n)l=\inf_{f\in C^\infty(T^d_{R_n})}\int_{T^d_{R_n}}|l-\nabla f(x)|^2 e^{-2 V_0^n(x)}dx\Big/\int_{T^d_{R_n}} e^{-2 V_0^n(x)}dx $$ We obtain quantitative estimates on $D(V_0^n)$, putting into evidence its geometric rate of convergence towards 0. From this we deduce the anomalous slow behavior of solutions of $dy_t=d\omega_t -\nabla V_0^\infty(y_t) dt$ using the tools of homogenization.