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Spectral properties of fractional Fokker-Plank operator for the L\'evy flight in a harmonic potential

Research paper by Ralf Toenjes, Igor M. Sokolov, Eugene B. Postnikov

Indexed on: 08 Oct '14Published on: 08 Oct '14Published in: Mathematical Physics



Abstract

We present a detailed analysis of the eigenfunctions of the Fokker-Planck operator for the L\'evy-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 L\'evy-Ornstein-Uhlenbeck process and their correlation functions.