An effective criterion and a new example for ballistic diffusions in random environment

Abstract

In the setting of multidimensional diffusions in random environment, we carry on the investigation of condition $(T^{\prime })$, introduced by Sznitman [Ann. Probab. 29 (2001) 723--764] and by Schmitz [Ann. Inst. H. Poincaré Probab. Statist. 42 (2006) 683--714] respectively in the discrete and continuous setting, and which implies a law of large numbers with nonvanishing limiting velocity (ballistic behavior) as well as a central limit theorem. Specifically, we show that when $d\ge 2$, $(T^{\prime })$ is equivalent to an effective condition that can be checked by local inspection of the environment. When $d=1$, we prove that condition $(T^{\prime })$ is merely equivalent to almost sure transience. As an application of the effective criterion, we show that when $d\ge 4$ a perturbation of Brownian motion by a random drift of size at most $\epsilon >0$ whose projection on some direction has expectation bigger than ${\epsilon }^{2-\eta },\eta >0$, satisfies condition $(T^{\prime })$ when $\epsilon$ is small and hence exhibits ballistic behavior. This class of diffusions contains new examples of ballistic behavior which in particular do not fulfill the condition in [Ann. Inst. H. Poincaré Probab. Statist. 42 (2006) 683--714], (5.4) therein, related to Kalikow's condition.

Published in at http://dx.doi.org/10.1214/07-AOP354 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)