Open Access
July, 1988 Nash Estimates and the Asymptotic Behavior of Diffusions
K. Golden, S. Goldstein, J. L. Lebowitz
Ann. Probab. 16(3): 1127-1146 (July, 1988). DOI: 10.1214/aop/1176991680


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 $B = 0$.


Download Citation

K. Golden. S. Goldstein. J. L. Lebowitz. "Nash Estimates and the Asymptotic Behavior of Diffusions." Ann. Probab. 16 (3) 1127 - 1146, July, 1988.


Published: July, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0648.60080
MathSciNet: MR942758
Digital Object Identifier: 10.1214/aop/1176991680

Primary: 60J60
Secondary: 35K10 , 82A42

Keywords: diffusion in a potential , invariance principle , local perturbations , Nash estimates , velocity autocorrelation function

Rights: Copyright © 1988 Institute of Mathematical Statistics

Vol.16 • No. 3 • July, 1988
Back to Top