We consider a transient diffusion in a (−κ/2)-drifted Brownian potential Wκ with 0 < κ < 1. We prove its localization at time t in the neighborhood of some random points depending only on the environment, which are the positive ht-minima of the environment, for ht a bit smaller than log t. We also prove an Aging phenomenon for the diffusion, a renewal theorem for the hitting time of the farthest visited valley, and provide a central limit theorem for the number of valleys visited up to time t.The proof relies on a decomposition of the trajectory of Wκ in the neighborhood of htminima, with the help of results of Faggionato [25], and on a precise analysis of exponential functionals of Wκ and of Wκ Doob-conditioned to stay positive. P(.) := P Wκ (.)P (W κ ∈ dω).We denote respectively by E Wκ , E, and E the expectations with regard to P Wκ , P and P .Date: 08/11/2018. 2010 Mathematics Subject Classification. 60F05, 60K05, 60K37, 60J60, 82D30.

We study a one-dimensional diffusion X in a drifted Brownian potential W κ , with 0 < κ < 1, and focus on the behavior of the local times (L(t, x), x) of X before time t > 0. In particular we characterize the limit law of the supremum of the local time, as well as the position of the favorite site. These limits can be written explicitly from a two dimensional stable Lévy process. Our analysis is based on the study of an extension of the renewal structure which is deeply involved in the asymptotic behavior of X.

We consider a branching system of random walks in random environment in Z, for which extinction is possible. We study the speed of the rightmost particle, conditionally on the survival of the branching process.

We consider a model of random walk in Z 2 with (fixed or random) orientation of the horizontal lines (layers) and with non constant iid probability to stay on these lines. We prove the transience of the walk for any fixed orientations under general hypotheses. This contrasts with the model of Campanino and Petritis [3], in which probabilities to stay on these lines are all equal. We also establish a result of convergence in distribution for this walk with suitable normalizations under more precise assumptions. In particular, our model proves to be, in many cases, even more superdiffusive than the random walks introduced by Campanino and Petritis. Layer k Layer kWe also define the annealed probability P as follows: P(.) := P ε,ω (.)dη(ω) dµ(ε).Date: November 1, 2018. 2010 Mathematics Subject Classification. 60F17; 60G52; 60K37. Key words and phrases. random walk on randomly oriented lattices, random walk in random environment, random walk in random scenery, functional limit theorem, transience. This research was supported by the french ANR project MEMEMO2 2010 BLAN 0125.