Nash estimates and the asymptotic behavior of diffusions

In order to analyze the asymptotic behavior of a particle diffusing in a drift field derived from a smooth bounded potential, we develop Nash-type a priori estimates on the transition density of the process. As an immediate consequence of the estimates, we find that for a rapidly decaying potential in Rd, the mean squared displacement behaves like td + C(t), where C(t) (the time integral of the "velocity autocorrelation function") decays like t-d/2. We also prove, using the estimates, that for a potential in Rd of the form V + B, where V is stationary random ergodic and B has compact support, the diffusion converges under space and time scaling to the same Brownian motion as does the diffusion with B = 0.