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

Abstract

We consider a transient diffusion in a $(-\kappa /2)$-drifted Brownian potential $W\_\kappa$ with $0\textless{}κ\textless{}1$. We prove its localization at time $t$ in the neighborhood of some random points depending only on the environment, which are the positive $h\_t$-minima of the environment, for $h\_t$ 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 $W\_\kappa$ in the neighborhood of $h\_t$-minima, with the help of results of A. Faggionato, and on a precise analysis of exponential functionals of $W\_\kappa$ and of $W\_\kappa$ Doob-conditioned to stay positive.

55 pages, 2 figures

References in corpus (2)

• Almost sure asymptotics for a diffusion process in a drifted Brownian potential
• Persistence of some additive functionals of Sinai's walk

Cited by in corpus (1)

• Persistence of some additive functionals of Sinai's walk