A Central Limit Theorem for Diffusions with Periodic Coefficients

Abstract

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.