Anomalous slow diffusion from perpetual homogenization

This paper is concerned with the asymptotic behavior of solutions of stochastic differential equations dy(t) = domega(t) - delV(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) = Sigma(k=0)(infinity) 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 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 greater than or equal to 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.