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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. https://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
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/math/0105165arXivDiscussion Paper
https://doi.org/10.1214/aop/1068646372DOIUNSPECIFIED
https://doi.org/10.1214/aop/1068646372DOIUNSPECIFIED
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