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November 1999 Multiscale diffusion processes with periodic coefficients and an application to solute transport in porous media
Rabi Bhattacharya
Ann. Appl. Probab. 9(4): 951-1020 (November 1999). DOI: 10.1214/aoap/1029962863

Abstract

Consider diffusions on $\mathbb{R}^k > 1$, governed by the Itô equation $dX(t) = {b(X(t)) + \beta(X(t)/a)} dt + \sigmadB(t)$, where $b, \beta$ are periodic with the same period and are divergence free, $\sigma$ is nonsingular and a is a large integer. Two distinct Gaussian phases occur as time progresses. The initial phase is exhibited over times $1 \ll t \ll a^{2/3}$. Under a geometric condition on the velocity field $\beta$, the final Gaussian phase occurs for times $t \gg a^2(\log a)^2$, and the dispersion grows quadratically with a . Under a complementary condition, the final phase shows up at times $t \gg a^4(\log a)^2$, or $t \gg a^2 \log a$ under additional conditions, with no unbounded growth in dispersion as a function of scale. Examples show the existence of non-Gaussian intermediate phases. These probabilisitic results are applied to analyze a multiscale Fokker-Planck equation governing solute transport in periodic porous media. In case $b, \beta$ are not divergence free, some insight is provided by the analysis of one-dimensional multiscale diffusions with periodic coefficients.

Citation

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Rabi Bhattacharya. "Multiscale diffusion processes with periodic coefficients and an application to solute transport in porous media." Ann. Appl. Probab. 9 (4) 951 - 1020, November 1999. https://doi.org/10.1214/aoap/1029962863

Information

Published: November 1999
First available in Project Euclid: 21 August 2002

zbMATH: 0956.60080
MathSciNet: MR1727912
Digital Object Identifier: 10.1214/aoap/1029962863

Subjects:
Primary: 60F60, 60J05
Secondary: 60H10, 60J70

Rights: Copyright © 1999 Institute of Mathematical Statistics

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Vol.9 • No. 4 • November 1999
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