Unusual eigenvalue spectrum and relaxation in the Lévy–Ornstein-Uhlenbeck process

We consider the rates of relaxation of a particle in a harmonic well, subject to Lévy noise characterized by its Lévy index $\mu$. Using the propagator for this Lévy–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 nondegenerate, but rational $\mu$ can lead to degeneracy. The maximum degeneracy is shown to be 2. 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 ${\left|x\right|}^{-n_1-n_2\mu }$ as $\left|x\right|\to \infty$, 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 can have nonspectral relaxation, as pointed out by $et$ $al$. [Phys. Rev. Lett. 110, 150602 (2013)].

Figure 1

Eigenvalues $n+m\mu$ plotted as functions of $\mu$ for $\mu \in [0,2]$, for $n=0,1,2...6$ and $m=0,1,2...6$. The spectrum has only nondegenerate eigenvalues if $\mu$ is irrational. Intersections of two or more lines occur when $\mu$ is rational and can lead to points where there is degeneracy. However, the maximum degeneracy is only 2, even though more than two lines may intersect at the same point. At $\mu =2,$ all eigenvalues are nondegenerate, and when $\mu =1,$ all eigenvalues except the lowest are doubly degenerate.