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

作者：Hoegele Michael; Pavlyukevich Ilya

来源：Stochastic Analysis and Applications, 2014, 32(1): 163-190.

DOI：10.1080/07362994.2014.858554

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

We consider a finite-dimensional deterministic dynamical system with the global attractor ? 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 ?. We perturb the dynamical system by a multiplicative heavy tailed Levy noise of small intensity E%26gt;0 and solve the asymptotic first exit time and location problem from D in the limit of E?0. In contrast to the case of Gaussian perturbations, the exit time has an algebraic exit rate as a function of E, just as in the case when ? 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 -stable Levy noise.

• 出版日期2014-1-2