Examples of Condition $(T)$ for Diffusions in a Random Environment

Abstract

With the help of the methods developed in our previous article [Schmitz, to appear in Annales de l'I.H.P., in press], we highlight condition $(T)$ as a source of new examples of 'ballistic' diffusions in a random environment when $d>1$ ('ballistic' means that a strong law of large numbers with non-vanishing limiting velocity holds). In particular we are able to treat the case of non-constant diffusion coefficients, a feature that causes problems. Further we recover the ballistic character of two important classes of diffusions in a random environment by simply checking condition $(T)$. This not only points out to the broad range of examples where condition $(T)$ can be checked, but also fortifies our belief that condition $(T)$ is a natural contender for the characterisation of ballistic diffusions in a random environment when $d>1$.