Diffusions in random environment and ballistic behavior

This article is accepted for publication in the "Annals I.H.P. Prob. & Stat.". 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. Sznitman \cite{szn01}, \cite{szn02}, that imply in higher dimensions a strong law of large numbers with non-vanishing limiting velocity (which we refer to as 'ballistic behavior') and a functional 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 new examples of ballistic diffusions in random environment.