Diffusions in random environment and ballistic behavior

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 bo...

Sharp ellipticity conditions for ballistic behavior of random walks in random environment

Sub-ballistic random walk in Dirichlet environment

Abstract. We consider random walks in Dirichlet environment (RWDE) on Z, for d > 3, in the sub-ballistic case. We associate to any parameter (α1, . . . , α2d) of the Dirichlet law a time-change to accelerate the walk. We prove that the continuoustime accelerated walk has an absolutely continuous invariant probability measure for the environment viewed from the particle. This allows to character...

Ballistic Random Walks in Random Environment at Low Disorder

We consider random walks in a random environment of the type p0 + γξz, where p0 denotes the transition probabilities of a stationary random walk on Z, to nearest neighbors, and ξz is an i.i.d. random perturbation. We give an explicit expansion, for small γ, of the asymptotic speed of the random walk under the annealed law, up to order 2. As an application, we construct, in dimension d ≥ 2, a wa...

Slowdown Estimates for Ballistic Random Walk in Random Environment

We consider random walk in elliptic i.i.d. random environment in dimension greater than or equal to 4, satisfying the bal-listicity condition (T ′). We show that for every ǫ > 0 and n large enough, the annealed probability of linear slowdown is bounded above by exp`− (log n) d−ǫ´. This is almost matching the known lower bound of exp`− C(log n) d ´ , and significantly improves previously known u...