Cut Points and Diffusions in Random Environment

TL;DR

This paper studies the long-term behavior of multi-dimensional diffusions in random environments, introducing cut times to decouple the process and establish laws of large numbers and central limit theorems.

Contribution

It introduces a novel approach using cut times in continuous diffusions to analyze their asymptotic behavior in random environments.

Findings

Established a strong law of large numbers for the diffusion process.

Proved a central limit theorem under the quenched measure.

Demonstrated the effectiveness of cut times in decoupling complex stochastic processes.

Abstract

In this article we investigate the asymptotic behavior of a new class of multi-dimensional diffusions in random environment. We introduce cut times in the spirit of the work done by Bolthausen, Sznitman and Zeitouni, see [4], in the discrete setting providing a decoupling effect in the process. This allows us to take advantage of an ergodic structure to derive a strong law of large numbers with possibly vanishing limiting velocity and a central limit theorem under the quenched measure.