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# Anomalous slow diffusion from perpetual homogenization

Owhadi, Houman (2003) Anomalous slow diffusion from perpetual homogenization. Annals of Probability, 31 (4). pp. 1935-1969. ISSN 0091-1798. http://resolver.caltech.edu/CaltechAUTHORS:OWHaop03

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## Abstract

This paper is concerned with the asymptotic behavior of solutions of stochastic differential equations $dy_t=d\omega_t -\nabla V(y_t)\, dt$, $y_0=0$. 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 subharmonic 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' 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.

Item Type: Article 2003 © Institute of Mathematical Statistics. Received March 2001; revised July 2002. This research was done at the EPFL in Lausanne. The author would like to thank Gé Ben Arous for stimulating discussions; the idea to investigate the link between the slow behavior of a Brownian motion and the presence of an infinite number of scales of obstacle comes from his work in geology, and the work of M. Barlow and R. Bass on the Sierpinski carpet. Thanks are also due to Hamish Short and to the referee for carefully reading the manuscript and for providing many useful comments. Multi-scale homogenization; anomalous diffusion; diffusion on fractal media; heat kernel; subharmonic; exponential martingale inequality; Davies' conjecture; periodic operator CaltechAUTHORS:OWHaop03 http://resolver.caltech.edu/CaltechAUTHORS:OWHaop03 http://dx.doi.org/10.1214/aop/1068646372 No commercial reproduction, distribution, display or performance rights in this work are provided. 9058 CaltechAUTHORS Archive Administrator 24 Oct 2007 26 Dec 2012 09:45

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