On the uniform convergence of the empirical density of an ergodic diffusion

We investigate the uniform convergence of the density of the empirical measure of an ergodic diffusion. It is known that under certain conditions on the drift and diffusion coefficients of the diffusion, the empirical density f t converges in probability to the invariant density f, uniformly on the entire real line. We show that under the same conditions, uniform convergence of f t to f on compact intervals takes place almost surely. Moreover, we prove that under much milder conditions (the usual linear growth condition on the drift and diffusion coefficients and a finite second moment of the invariant measure suffice), we have the uniform convergence of f t to f on compacta in probability.