On stochastic perturbations of slowly changing dynamical systems

TL;DR

This paper analyzes how small random perturbations affect the behavior of dynamical systems with slowly changing parameters, focusing on exit times and distribution of exit points.

Contribution

It provides asymptotic descriptions of exit times and distributions for diffusion processes with slowly evolving drift and diffusion terms.

Findings

Asymptotic formulas for exit times from a domain.

Limiting distribution of exit points.

Effects of slow parameter changes on stochastic stability.

Abstract

In this paper we consider a diffusion process obtained as a small random perturbation of a dynamical system attracted to a stable equilibrium point. The drift and the diffusive perturbation are assumed to evolve slowly in time. We describe the asymptotics of the time it takes the process to exit a given domain and the limiting distribution of the exit point.