Anomalous Slow Diffusion from Perpetual Homogenization

Abstract

This paper is concerned with the asymptotic behavior of solutions of stochastic differential equations $dy_t=dω_t -\nabla V(y_t) dt$, $y_0=0$. When $d=1$ and $V$ is not periodic but obtained as a superposition of an infinite number of periodic potentials with geometrically increasing periods ($V(x) = \sum_{k=0}^\infty U_k(x/R_k)$, where $U_k$ are smooth functions of period 1, $U_k(0)=0$, and $R_k$ grows exponentially fast with $k$) we can show that $y_t$ has an anomalous slow behavior and we obtain quantitative estimates on the anomaly using and developing the tools of homogenization. Pointwise estimates are based on a new analytical inequality for sub-harmonic functions. When $d\geq 1$ and $V$ is periodic, quantitative estimates are obtained on the heat kernel of $y_t$, showing the rate at which homogenization takes place. The latter result proves Davies's conjecture and is based on a quantitative estimate for the Laplace transform of martingales that can be used to obtain similar results for periodic elliptic generators