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.

Keywords:

• α-stable Lévy process
• Canonical (Marcus) SDE
• First exit location
• First exit time
• Global attractor
• Itô SDE
• Multiplicative noise
• Regular variation
• Stratonovich SDE
• Van der Pol oscillator

2010 Mathematics Subject Classification:

• 60H10
• 60G51
• 37A20
• 60J60
• 60J75
• 60G52

Acknowledgments

M. Högele and I. Pavlyukevich would like to express his gratitude to the Berlin Mathematical School (BMS), International Research Training Group (IRTG) 1740: “Dynamical Phenomena in Complex Networks: Fundamentals and Applications” and the Stochastics group at Humboldt-Universität zu Berlin for various infrastructure support. Michael Högele and I. Both authors would like to thank ZiF Bielefeld for the hospitality during the workshop of the Cooperation Group, “Exploring Climate Variability: Physical Models, Statistical Inference and Stochastic Dynamics’ (February 18 to March 28, 2013), during which this work was finished. The authors are grateful to the anonymous referee for the careful reading and helpful suggestions, which have led to a significant improvement of the manuscript.