Eur. Phys. J. B (2014) 87: 287
https://doi.org/10.1140/epjb/e2014-50558-5

Regular Article

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

¹ Institute of Physics and Astronomy, Potsdam University, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany
² Institut für Physik, Humboldt Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
³ Department of Theoretical Physics, Kursk State University, Radishcheva st., 33, 305000 Kursk, Russia

^a e-mail: toenjes@uni-potsdam.de

Received: 18 August 2014
Received in final form: 27 October 2014
Published online: 3 December 2014

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.