Unusual eigenvalue spectrum and relaxation in the L\'{e}vy Ornstein-Uhlenbeck process

Abstract

We consider the rates of relaxation of a particle in a harmonic well, subject to L\'evy noise characterized by its L\'evy index $\mu$. Using the propagator for this L\'evy Ornstein-Uhlenbeck process (LOUP), we show that the eigenvalue spectrum of the associated Fokker-Planck operator has the form $(n+m\mu)\nu$ where $\nu$ is the force constant characterizing the well, and $n,m\in\mathbb{N}$. If $\mu$ is irrational, the eigenvalues are all non-degenerate, but rational $\mu$ can lead to degeneracy. The maximum degeneracy is shown to be two. The left eigenfunctions of the fractional Fokker-Planck operator are very simple while the right eigenfunctions may be obtained from the lowest eigenfunction by a combination of two different step-up operators. Further, we find that the acceptable eigenfunctions should have the asymptotic behavior $|x|^{-n_1+n_2\;\mu}$ as $|x| \rightarrow \infty$, with $n_1$ and $n_2$ being positive integers, though this condition alone is not enough to identify them uniquely. We also assert that the rates of relaxation of LOUP are determined by the eigenvalues of the associated fractional Fokker-Planck operator and do not depend on the initial state if the moments of the initial distribution are all finite. If the initial distribution has fat tails, for which the higher moments diverge, one would have non-spectral relaxation, as pointed out by Toenjes et. al (Physical Review Letters, 110, 150602 (2013)).