Gaussian limit theorems for diffusion processes and an application

Suppose that L=∑i, j=1daij(x)∂2/∂xi∂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 {x:[aij(x)]≠12I} is strictly less than d, the random variable (XL(T)−XL(0))/T 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.