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

摘要

We consider the rates of relaxation of a particle in a harmonic well, subjectto L\'evy noise characterized by its L\'evy index $\mu$. Using the propagatorfor this L\'evy Ornstein-Uhlenbeck process (LOUP), we show that the eigenvaluespectrum 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 allnon-degenerate, but rational $\mu$ can lead to degeneracy. The maximumdegeneracy is shown to be two. The left eigenfunctions of the fractionalFokker-Planck operator are very simple while the right eigenfunctions may beobtained from the lowest eigenfunction by a combination of two differentstep-up operators. Further, we find that the acceptable eigenfunctions shouldhave 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 isnot enough to identify them uniquely. We also assert that the rates ofrelaxation of LOUP are determined by the eigenvalues of the associatedfractional Fokker-Planck operator and do not depend on the initial state if themoments of the initial distribution are all finite. If the initial distributionhas fat tails, for which the higher moments diverge, one would havenon-spectral relaxation, as pointed out by Toenjes et. al (Physical ReviewLetters, 110, 150602 (2013)).