The Annals of Probability

A Central Limit Theorem for Diffusions with Periodic Coefficients

Rabi Bhattacharya

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It is proved that if $X_t$ is a diffusion generated by the operator $L = 1/2 \sum a_{ij}(x)\partial^2/\partial x_i\partial x_j + \sum u_0b_i(x)\partial/\partial x_i$ having periodic coefficients, then $\lambda^{-1/2}(X_{\lambda t} - \lambda u_0 \bar{b}t), t \geq 0$, converges in distribution to a Brownian motion as $\lambda \rightarrow \infty$. Here $\bar{b}$ is the mean of $b(x) = (b_1(x), \cdots, b_k(x))$ with respect to the invariant distribution for the diffusion induced on the torus $T^k = \lbrack 0, 1)^k$. The dispersion matrix of the limiting Brownian motion is also computed. In case $\bar{b} = 0$ this result was obtained by Bensoussan, Lions and Papanicolaou (1978). (See Theorem 4.3, page 401, as well as the author's remarks on page 529.) The case $\bar{b} \neq 0$ is of interest in understanding how solute dispersion in a porous medium behaves as the liquid velocity increases in magnitude.

Article information

Ann. Probab., Volume 13, Number 2 (1985), 385-396.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60F17: Functional limit theorems; invariance principles

Markov process on the torus generator dispersion


Bhattacharya, Rabi. A Central Limit Theorem for Diffusions with Periodic Coefficients. Ann. Probab. 13 (1985), no. 2, 385--396. doi:10.1214/aop/1176992998.

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