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

TL;DR

This paper investigates the condition (T') for multidimensional diffusions in random environments, establishing an effective criterion for checking ballistic behavior and providing new examples of such diffusions.

Contribution

It introduces an effective local inspection criterion for condition (T') in dimensions d≥2 and applies it to demonstrate ballistic behavior in perturbed Brownian motions.

Findings

Condition (T') is equivalent to an easily checkable local condition for d≥2.

For d=1, (T') is equivalent to almost sure transience.

Small perturbations of Brownian motion can satisfy (T') and exhibit ballistic behavior.

Abstract

In the setting of multidimensional diffusions in random environment, we carry on the investigation of condition $(T')$, introduced by Sznitman [Ann. Probab. 29 (2001) 723--764] and by Schmitz [Ann. Inst. H. Poincar\'{e} 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 $d\geq2$, $(T')$ is equivalent to an effective condition that can be checked by local inspection of the environment. When $d=1$, we prove that condition $(T')$ is merely equivalent to almost sure transience. As an application of the effective criterion, we show that when $d\geq4$ 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 $(T')$ 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. Poincar\'{e} Probab. Statist. 42 (2006) 683--714], (5.4) therein, related to Kalikow's condition.