ANOMALOUS SLOW DIFFUSION FROM PERPETUAL HOMOGENIZATION BY HOUMAN OWHADI Universite de Provence

This paper is concerned with the asymptotic behavior of solutions of stochastic differential equations dyt = dωt − ∇V (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)= ∑∞ 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 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 yt , 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.