The Exit Problem from a Neighborhood of the Global Attractor for Dynamical Systems Perturbed by Heavy-Tailed Lévy Processes

Abstract

We consider a finite-dimensional deterministic dynamical system with the global attractor 𝒜 which supports a unique ergodic probability measure P. The measure P can be considered as the uniform long-term mean of the trajectories staying in a bounded domain D containing 𝒜. We perturb the dynamical system by a multiplicative heavy tailed Lévy noise of small intensity ϵ > 0 and solve the asymptotic first exit time and location problem from D in the limit of ϵ↘0. In contrast to the case of Gaussian perturbations, the exit time has an algebraic exit rate as a function of ϵ, just as in the case when 𝒜 is a stable fixed point studied earlier in [9, 14, 19, 26]. As an example, we study the first exit problem from a neighborhood of the stable limit cycle for the Van der Pol oscillator perturbed by multiplicative α-stable Lévy noise.