Gaussian limit theorems for diffusion processes and an application

Suppose that L=[summation operator]i, j=1daij(x)[not partial differential]2/[not partial differential]xi[not partial differential]xj is uniformly elliptic. We use XL(t) to denote the diffusion associated with L. In this paper we show that, if the dimension of the set is strictly less than d, the random variable converges in distribution to a standard Gaussian random variable. In fact, we also provide rates of convergence. As an application, these results are used to study a problem of a random walk in a random environment.