Diffusions in random environment and ballistic behavior

Abstract

In this article we investigate the ballistic behavior of diffusions in random environment. We introduce conditions in the spirit of ( T ) and ( T ′ ) of the discrete setting, cf. [A.-S. Sznitman, On a class of transient random walks in random environment, Ann. Probab. 29 (2) (2001) 723–764; A.-S. Sznitman, An effective criterion for ballistic behavior of random walks in random environment, Probab. Theory Related Fields 122 (4) (2002) 509–544], that imply, when d ⩾ 2 , a law of large numbers with non-vanishing limiting velocity (which we refer to as ‘ballistic behavior’) and a central limit theorem with non-degenerate covariance matrix. As an application of our results, we consider the class of diffusions where the diffusion matrix is the identity, and give a concrete criterion on the drift term under which the diffusion in random environment exhibits ballistic behavior. This criterion provides examples of diffusions in random environment with ballistic behavior, beyond what was previously known.