Scaling Limits and Exit Law for Multiscale Diffusions

TL;DR

This paper investigates the behavior of multiscale diffusions under small noise perturbations, focusing on fluctuations, exit times, and locations, with applications to Langevin dynamics in rough potentials.

Contribution

It provides new results on the limiting laws of fluctuations and exit distributions for multiscale diffusions, including Langevin equations in complex potentials.

Findings

Derived the limiting law of joint exit time and location

Analyzed fluctuations around typical behavior in multiscale diffusions

Studied rare event exit laws for Langevin equations in rough potentials

Abstract

In this paper we study the fluctuations from the limiting behavior of small noise random perturbations of diffusions with multiple scales. The result is then applied to the exit problem for multiscale diffusions, deriving the limiting law of the joint distribution of the exit time and exit location. We apply our results to the first order Langevin equation in a rough potential, studying both fluctuations around the typical behavior and the conditional limiting exit law, conditional on the rare event of going against the underlying deterministic flow.