Metastable behaviour of small noise Lévy-Driven diffusions

Issue		ESAIM: PS Volume 12


Page(s)		412 - 437
DOI		10.1051/ps:2007051
Published online		25 July 2008

ESAIM: PS, July 2008, Vol. 12, p. 412-437
DOI: 10.1051/ps:2007051

Metastable behaviour of small noise Lévy-Driven diffusions

Peter Imkeller and Ilya Pavlyukevich

Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany; imkeller@mathematik.hu-berlin.de; pavljuke@mathematik.hu-berlin.de

Received September 12, 2007. Published online 25 July 2008

Abstract
We consider a dynamical system in $\mathbb{R}$ driven by a vector field -U', where U is a multi-well potential satisfying some regularity conditions. We perturb this dynamical system by a Lévy noise of small intensity and such that the heaviest tail of its Lévy measure is regularly varying. We show that the perturbed dynamical system exhibits metastable behaviour i.e. on a proper time scale it reminds of a Markov jump process taking values in the local minima of the potential U. Due to the heavy-tail nature of the random perturbation, the results differ strongly from the well studied purely Gaussian case.

Mathematics Subject Classification. 60E07, 60F10

Key words: Lévy process, jump diffusion, heavy tail, regular variation, metastability, extreme events, first exit time, large deviations

© EDP Sciences, SMAI 2008

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