The exit problem from the neighborhood of a global attractor for heavy-tailed L\'evy diffusions

TL;DR

This paper analyzes the exit times and locations for a deterministic dynamical system perturbed by small heavy-tailed Lévy noise, revealing algebraic exit rates and joint distributions, with applications to the Van der Pol oscillator.

Contribution

It provides the first asymptotic analysis of exit times under heavy-tailed Lévy noise, contrasting with Gaussian cases, and characterizes the joint law of exit time and location.

Findings

Exit time has algebraic asymptotic rate in noise intensity.

Joint law of exit time and location is characterized.

Application to Van der Pol oscillator with Lévy noise.

Abstract

We consider a finite dimensional deterministic dynamical system with a global attractor A with a unique ergodic measure P concentrated on it, which is uniformly parametrized by the mean of the trajectories in a bounded set D containing A. We perturbe this dynamical system by a multiplicative heavy tailed L\'evy noise of small intensity \epsilon>0 and solve the asymptotic first exit time and location problem from a bounded domain D around the attractor A in the limit of {\epsilon} to 0. In contrast to the case of Gaussian perturbations, the exit time has the asymptotically algebraic exit rate as a function of \epsilon, just as in the case when A is a stable fixed point. In the small noise limit, we determine the joint law of the first time and the exit location on the complement of D. As an example, we study the first exit problem from a neighbourhood of a stable limit cycle for the Van der Pol oscillator perturbed by multiplicative \alpha-stable L\'evy noise.