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

TL;DR

This paper analyzes the spectral properties of the fractional Fokker-Planck operator for Lévy flights in a harmonic potential, providing explicit solutions and exploring non-spectral relaxation phenomena.

Contribution

It offers explicit eigenfunction expressions, asymptotic analysis, and a transformation kernel connecting fractional and non-fractional operators for Lévy-driven Ornstein-Uhlenbeck processes.

Findings

Explicit eigenfunctions for specific cases

Asymptotic behavior and recurrence relations derived

Identification of non-spectral relaxation in bounded Lévy processes

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.