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

With the help of the methods developed in [21], we highlight condition (T ) as a source of new examples of ’ballistic’ diffusions in a random environment when d ≥ 2 (’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 ≥ 2