The Annals of Applied Probability

Multiscale diffusion processes with periodic coefficients and an application to solute transport in porous media

Rabi Bhattacharya

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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.

Article information

Source
Ann. Appl. Probab., Volume 9, Number 4 (1999), 951-1020.

Dates
First available in Project Euclid: 21 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1029962863

Digital Object Identifier
doi:10.1214/aoap/1029962863

Mathematical Reviews number (MathSciNet)
MR1727912

Zentralblatt MATH identifier
0956.60080

Subjects
Primary: 60F60 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

Keywords
Diffusion on a big torus speed of convergence to equilibrium initial and final Gaussian phases growth in dispersion

Citation

Bhattacharya, Rabi. Multiscale diffusion processes with periodic coefficients and an application to solute transport in porous media. Ann. Appl. Probab. 9 (1999), no. 4, 951--1020. doi:10.1214/aoap/1029962863. https://projecteuclid.org/euclid.aoap/1029962863


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