Localization and number of visited valleys for a transient diffusion in random environment

Abstract

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.

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