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

In the setting of multidimensional diffusions in random environment, we carry on the investigation of condition $(T^{\mathrm{\prime }})$, 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 $d\ge 2$, $(T^{\mathrm{\prime }})$ is equivalent to an effective condition that can be checked by local inspection of the environment. When $d=1$, we prove that condition $(T^{\mathrm{\prime }})$ 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 $ϵ>0$ whose projection on some direction has expectation bigger than ${ϵ}^{2-\eta },\eta >0$, satisfies condition $(T^{\mathrm{\prime }})$ when $ϵ$ 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.