Estimates on the speedup and slowdown for a diffusion in a drifted brownian potential

TL;DR

This paper analyzes the deviations in the speed of diffusion within a Brownian potential, providing estimates on the process's speedup or slowdown relative to its typical behavior, using advanced probabilistic tools.

Contribution

It offers new estimates on the deviations of diffusion in a Brownian potential, extending discrete case results to continuous models with novel analytical techniques.

Findings

Quantitative estimates of diffusion speed deviations

Agreement with discrete case results

Application of advanced probabilistic tools

Abstract

We study a model of diffusion in a brownian potential. This model was firstly introduced by T. Brox (1986) as a continuous time analogue of random walk in random environment. We estimate the deviations of this process above or under its typical behavior. Our results rely on different tools such as a representation introduced by Y. Hu, Z. Shi and M. Yor, Kotani's lemma, introduced at first by K. Kawazu and H. Tanaka (1997), and a decomposition of hitting times developed in a recent article by A. Fribergh, N. Gantert and S. Popov (2008). Our results are in agreement with their results in the discrete case.