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 $0 < \kappa < 1$. 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 adecomposition 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.;