Asymptotic methods for stochastic dynamical systems with small non-Gaussian L\'evy noise

TL;DR

This paper develops asymptotic techniques to analytically approximate escape probabilities in stochastic dynamical systems driven by small, non-Gaussian symmetric alpha-stable Lévy noise, validated through examples and simulations.

Contribution

It introduces asymptotic methods for solving integro-differential equations related to escape probabilities under Lévy noise, a novel approach for such non-Gaussian systems.

Findings

Asymptotic escape probabilities closely match numerical simulations.

Methods effectively handle nonlocal integro-differential equations.

Illustrative examples demonstrate practical applicability.

Abstract

The goal of the paper is to analytically examine escape probabilities for dynamical systems driven by symmetric $\alpha$-stable L\'evy motions. Since escape probabilities are solutions of a type of integro-differential equations (i.e., differential equations with nonlocal interactions), asymptotic methods are offered to solve these equations to obtain escape probabilities when noises are sufficiently small. Three examples are presented to illustrate the asymptotic methods, and asymptotic escape probability is compared with numerical simulations.