## The Annals of Applied Probability

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

Rabi Bhattacharya

#### 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

https://projecteuclid.org/euclid.aoap/1029962863

Digital Object Identifier
doi:10.1214/aoap/1029962863

Mathematical Reviews number (MathSciNet)
MR1727912

Zentralblatt MATH identifier
0956.60080

#### 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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