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. Prob. & Stat.], 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.