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October 2003 Anomalous slow diffusion from perpetual homogenization
Houman Owhadi
Ann. Probab. 31(4): 1935-1969 (October 2003). DOI: 10.1214/aop/1068646372


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.


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Houman Owhadi. "Anomalous slow diffusion from perpetual homogenization." Ann. Probab. 31 (4) 1935 - 1969, October 2003.


Published: October 2003
First available in Project Euclid: 12 November 2003

zbMATH: 1042.60049
MathSciNet: MR2016606
Digital Object Identifier: 10.1214/aop/1068646372

Primary: 60J60
Secondary: 31C05 , 34E13 , 35B27 , 60F05 , 60G44

Keywords: Anomalous diffusion , Davies' conjecture , diffusion on fractal media , exponential martingale inequality , heat kernel , Multi-scale homogenization , periodic operator , subharmonic

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 4 • October 2003
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