Owhadi, Houman (2003) Anomalous slow diffusion from perpetual homogenization. Annals of Probability, 31 (4). pp. 19351969. ISSN 00911798. 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 

Additional Information:  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. 
Subject Keywords:  Multiscale homogenization; anomalous diffusion; diffusion on fractal media; heat kernel; subharmonic; exponential martingale inequality; Davies' conjecture; periodic operator 
Record Number:  CaltechAUTHORS:OWHaop03 
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:OWHaop03 
Alternative URL:  http://dx.doi.org/10.1214/aop/1068646372 
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ID Code:  9058 
Collection:  CaltechAUTHORS 
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Deposited On:  24 Oct 2007 
Last Modified:  26 Dec 2012 09:45 
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