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 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 Lévy noise of small intensity ε > 0 and solve the asymptotic first exit time and location problem from a bounded domain D around the attractor A in the limit of ε ց 0. In contrast to the case of Gaussian perturbations, the exit time has the asymptotically algebraic exit rate as a function of ε, just as in the case when A is a stable fixed point (see for instance [9,18,24]). In the small noise limit, we determine the joint law of the first time and the exit location from D c . 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 α-stable Lévy noise.

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