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First Exit Times of Non-linear Dynamical Systems in ℝd Perturbed by Multifractal Lévy Noise

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Abstract

In a domain \(\mathcal{G}\subset\mathbb{R}^{d}\) we study a dynamical system which is perturbed in finitely many directions i by one-dimensional Lévy processes with α i -stable components. We investigate the exit behavior of the system from the domain in the small noise limit. Using probabilistic estimates on the Laplace transform of the exit time we show that it is exponentially distributed with a parameter that depends on the smallest α i . Finally we prove that the system exits from the domain in the direction of the process with the smallest α i .

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Authors and Affiliations

  1. Institut für Mathematik, Humboldt–Universität zu Berlin, Rudower Chaussee 25, 12489, Berlin, Germany

    Peter Imkeller & Michael Stauch

  2. Institut für Stochastik, Friedrich–Schiller–Universität Jena, Ernst–Abbe–Platz 2, 07743, Jena, Germany

    Ilya Pavlyukevich

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  1. Peter Imkeller
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  2. Ilya Pavlyukevich
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  3. Michael Stauch
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Correspondence to Ilya Pavlyukevich.

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Imkeller, P., Pavlyukevich, I. & Stauch, M. First Exit Times of Non-linear Dynamical Systems in ℝd Perturbed by Multifractal Lévy Noise. J Stat Phys 141, 94–119 (2010). https://doi.org/10.1007/s10955-010-0041-6

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  • DOI: https://doi.org/10.1007/s10955-010-0041-6

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