Anomalous Slow Diffusion from Perpetual Homogenization

This paper is concerned with the asymptotic behavior solutions of stochastic differential equations dyt = dωt − ∇V (yt)dt, y0 = 0. When d = 1 and V reflects media characterized by an infinite number of spatial scales V (x) = ∞ k=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 obtain quantitative estimates on the anomaly using and developing the tools of homogenisation. Pointwize estimates are based on a new analytical inequality for sub-harmonic functions. When d ≥ 1 and V is periodic, quantitative estimates are obtained on the heat kernel of yt, showing the rate at which homogenization takes place, the latter result is directly linked to Davies’s conjecture and based on a quantitative estimate for the Laplace transform of martingales that can be used to obtain similar results for periodic elliptic generator