Scaling Limit for the Diffusion Exit Problem in the Levinson Case

TL;DR

This paper investigates the asymptotic behavior of exit times and points for small random perturbations of dynamical systems under Levinson conditions, providing joint scaling limits and analyzing rare event scenarios.

Contribution

It introduces a joint scaling limit for exit time and point in perturbed dynamical systems satisfying Levinson conditions, extending understanding of exit problems under small stochastic influences.

Findings

Derived joint scaling limit for exit time and point.

Analyzed asymptotics of exit times conditioned on rare events.

Provided theoretical insights into diffusion exit problems in the Levinson case.

Abstract

The exit problem for small perturbations of a dynamical system in a domain is considered. It is assumed that the unperturbed dynamical system and the domain satisfy the Levinson conditions. We assume that the random perturbation affects the driving vector field and the initial condition, and each of the components of the perturbation follows a scaling limit. We derive the joint scaling limit for the random exit time and exit point. We use this result to study the asymptotics of the exit time for 1-d diffusions conditioned on rare events.