Nash Estimates and the Asymptotic Behavior of Diffusions

Abstract

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 $\mathbb{R}^d$, the mean squared displacement behaves like $td + C(t)$, where $\dot{C}(t)$ (the time integral of the "velocity autocorrelation function") decays like $t^{-d/2}$. We also prove, using the estimates, that for a potential in $\mathbb{R}^d$ 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 $B = 0$.