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

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.

arXiv comment10 pages, 5 figures, submitted to Euro. Phys. J. B

References in corpus (6)

• Fractional Quantum Mechanics
• Levy Flights in External Force Fields: Langevin and Fractional Fokker-Planck Equations, and their Solutions
• First Passage Time Densities in Resonate-and-Fire Models
• Levy flights and nonlocal quantum dynamics
• Non-spectral relaxation in one dimensional Ornstein-Uhlenbeck processes
• Unusual eigenvalue spectrum and relaxation in the Lévy Ornstein-Uhlenbeck process

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