We study diffusions in random environment in higher dimensions. We assume that the environment is stationary and obeys finite range dependence. Once the environment is chosen, it remains fixed in time. To restore some stationarity, it is common to average over the environment. One then obtains the so-called annealed measures, that are typically non-Markovian measures.
Our goal is to study the asymptotic behavior of the diffusion in random environment under the annealed measure, with particular emphasis on the ballistic regime
('ballistic' means that a law of large numbers with non-vanishing limiting velocity holds). In the spirit of Sznitman, who treated the discrete setting, we introduce
conditions (T) and (T'), and show that they imply, when d>1, a ballistic law of large numbers and a central limit theorem with non-degenerate covariance matrix.
As an application of our results, we highlight condition (T) as a source of new examples of ballistic diffusions in random environment.