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Spectral properties of the fractional Fokker-Planck operator for the Lévy flight in a harmonic potential

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Abstract

We present a detailed analysis of the eigenfunctions of the Fokker-Planck operator for the Lévy-Ornstein-Uhlenbeck process, their asymptotic behavior and recurrence relations, explicit expressions in coordinate space for the special cases of the Ornstein-Uhlenbeck process with Gaussian and with Cauchy white noise and for the transformation kernel, which maps the fractional Fokker-Planck operator of the Cauchy-Ornstein-Uhlenbeck process to the non-fractional Fokker-Planck operator of the usual Gaussian Ornstein-Uhlenbeck process. We also describe how non-spectral relaxation can be observed in bounded random variables of the Lévy-Ornstein-Uhlenbeck process and their correlation functions.

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Authors and Affiliations

  1. Institute of Physics and Astronomy, Potsdam University, Karl-Liebknecht-Strasse 24/25, 14476, Potsdam-Golm, Germany

    Ralf Toenjes

  2. Institut für Physik, Humboldt Universität zu Berlin, Newtonstraße 15, 12489, Berlin, Germany

    Igor M. Sokolov

  3. Department of Theoretical Physics, Kursk State University, Radishcheva st., 33, 305000, Kursk, Russia

    Eugene B. Postnikov

Authors
  1. Ralf Toenjes
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  2. Igor M. Sokolov
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  3. Eugene B. Postnikov
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Correspondence to Ralf Toenjes.

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Toenjes, R., Sokolov, I. & Postnikov, E. Spectral properties of the fractional Fokker-Planck operator for the Lévy flight in a harmonic potential. Eur. Phys. J. B 87, 287 (2014). https://doi.org/10.1140/epjb/e2014-50558-5

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  • DOI: https://doi.org/10.1140/epjb/e2014-50558-5

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