Escape Probability for Stochastic Dynamical Systems with Jumps

TL;DR

This paper studies the escape probabilities of stochastic dynamical systems driven by non-Gaussian Lévy motions, providing mathematical characterizations and highlighting differences from Gaussian-driven systems.

Contribution

It extends the analysis of escape probabilities to systems driven by non-Gaussian Lévy motions, offering new PDE-integral equation characterizations and analytic results.

Findings

Escape probabilities characterized by PDE-integral equations

Differences between Gaussian and non-Gaussian noise effects highlighted

Analytic solutions provided in special cases

Abstract

The escape probability is a deterministic concept that quantifies some aspects of stochastic dynamics. This issue has been investigated previously for dynamical systems driven by Gaussian Brownian motions. The present work considers escape probabilities for dynamical systems driven by non-Gaussian L\'evy motions, especially symmetric $\alpha$-stable L\'evy motions. The escape probabilities are characterized as solutions of the Balayage-Dirichlet problems of certain partial differential-integral equations. Differences between escape probabilities for dynamical systems driven by Gaussian and non-Gaussian noises are highlighted. In certain special cases, analytic results for escape probabilities are given.