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'), 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≥2, (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≥4 a perturbation of Brownian motion by a random drift of size at most ɛ>0 whose projection on some direction has expectation bigger than ɛ^2−η, η>0, satisfies condition (T') when ɛ 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.