Anomalous slow diffusion from perpetual homogenization

Abstract

This paper is concerned with the asymptotic behavior of solutions of stochastic differential equations dyt = dωt - ΔV (yt) dt, y0= 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) = Σk=0∞ Uk(x/Rk), where Uk are smooth functions of period 1, Uk(0) = 0, and Rk grows exponentially fast with k] we can show that yt 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 subharmonic functions. When d ≥ 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' 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.