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Phys. Rev. E 79, 011110 (2009) [5 pages]

Realization of Lévy walks as Markovian stochastic processes

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Ihor Lubashevsky1,2,3, Rudolf Friedrich4,5, and Andreas Heuer2,5
1A.M. Prokhorov General Physics Institute, Russian Academy of Sciences, Vavilov Strasse 38, 119991 Moscow, Russia
2Institut für Physikalische Chemie, Westfälische Wilhelms Universität Münster, Corrensstrasse 30, 48149 Münster, Germany
3Moscow Technical University of Radioengineering, Electronics, and Automation, Vernadsky 78, 119454, Moscow, Russia
4Institut für Theoretische Physik, Westfälische Wilhelms Universität Münster, Wilhelm-Klemm. 9, 48149 Münster, Germany
5Center of Nonlinear Science CeNoS, Westfälische Wilhelms Universität Münster, 48149 Münster, Germany

Received 28 June 2007; published 12 January 2009

Based on multivariate Langevin processes we present a realization of Lévy flights as a continuous process. For the simple case of a particle moving under the influence of friction and a velocity-dependent stochastic force we explicitly derive the generalized Langevin equation and the corresponding generalized Fokker-Planck equation describing Lévy flights. Our procedure is similar to the treatment of the Kramers-Fokker-Planck equation in the Smoluchowski limit. The proposed approach may open a way to treat Lévy flights in inhomogeneous media or systems with boundaries in the future.

© 2009 The American Physical Society

URL:
http://link.aps.org/doi/10.1103/PhysRevE.79.011110
DOI:
10.1103/PhysRevE.79.011110
PACS:
05.40.Fb, 02.50.Ey, 02.50.Ga, 05.10.Gg
 
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