Diffusions in random environment and ballistic behaviour

In this thesis, we study continuous diffusions in random environment on Rd . The randomness in the environment is incorporated in the diffusion matrix and the drift term that are stationary random variables. Once the environment is chosen, it remains fixed in time, and the diffusion evolving in this frozen environment is a Markov process. To restore some stationarity, it is common to average both with respect to the path and environment mesure. One then obtains the so-called annealed measures, under which the Markov property is typically lost. Our goal is to investigate the asymptotic behavior of the diffusion in random environment under the annealed measures, with particular emphasis on the ballistic regime (’ballistic’ means that a law of large numbers with nonvanishing 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 ≥ 2, a ballistic law of large numbers and a central limit theorem with non-degenerate covariance matrix. Our methods rely on the powerful renewal structure for diffusions in random environment provided by Shen. This renewal structure is induced by successive regeneration times τk , k ≥ 1. To show ballistic behavior amounts to derive good tail estimates on the first regeneration time τ1. As an application of our results, we point out to the broad range of examples where condition (T ) can be checked. When d ≥ 2, we not only recover the ballistic character of certain classes of diffusions in random environment previously obtained by different methods, but we also give a concrete criterion on the expectation of the drift term that provides new examples of ballistic diffusions in random environment.