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
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.
Key words: Statistical and Nonlinear Physics
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2014