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Metastable behaviour of small noise Lévy-Driven diffusions

Published online by Cambridge University Press:  25 July 2008

Peter Imkeller  and
Ilya Pavlyukevich
Peter Imkeller
Affiliation:
Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany; imkeller@mathematik.hu-berlin.de; pavljuke@mathematik.hu-berlin.de
Ilya Pavlyukevich
Affiliation:
Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany; imkeller@mathematik.hu-berlin.de; pavljuke@mathematik.hu-berlin.de
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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.

Keywords

Lévy processjump diffusionheavy tailregular variationmetastabilityextreme eventsfirst exit timelarge deviations
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 12 , 2008 , pp. 412 - 437
Copyright
© EDP Sciences, SMAI, 2008

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