Limit law of the local time for Brox's diffusion

TL;DR

This paper studies the asymptotic behavior of the local time for Brox's diffusion in a Brownian potential, revealing its convergence to a process related to Bessel processes and comparing it to discrete cases.

Contribution

It introduces the limit law of the normalized local time for Brox's diffusion and establishes its convergence to a functional of Bessel processes, extending understanding of diffusion in random environments.

Findings

Normalized local time converges to a process depending only on the potential W.

Weak convergence of local time to a functional of two independent 3D Bessel processes.

Limit law of the supremum of the normalized local time is derived.

Abstract

We consider Brox's model: a one-dimensional diffusion in a Brownian potential W. We show that the normalized local time process (L(t;m_(log t) + x)=t; x \in R), where m_(log t) is the bottom of the deepest valley reached by the process before time t, behaves asymptotically like a process which only depends on W. As a consequence, we get the weak convergence of the local time to a functional of two independent three-dimensional Bessel processes and thus the limit law of the supremum of the normalized local time. These results are discussed and compared to the discrete time and space case which same questions have been solved recently by N. Gantert, Y. Peres and Z. Shi.