Spectral theory for non-local Ornstein-Uhlenbeck operators

Abstract

We consider non-local Ornstein-Uhlenbeck (OU) operators generating ergodic OU processes driven by Lévy processes. These operators are generally non-normal in the $L^2$ space weighted by the invariant distribution. We prove that their point spectrum and eigenvalue multiplicities are independent of the driving Lévy process and are determined by the drift matrix. When the drift matrix is diagonalizable, we derive explicit eigenfunctions and co-eigenfunctions forming a biorthogonal system with respect to the invariant distribution. In the one-dimensional case, the eigefunctions form an \emph{Appell sequence}, which have appeared in the study of optimal stopping problems related to Lévy processes. We also study spectral expansion and compactness of the semigroup, and provide a class of compact non-local OU semigroups admitting a biorthogonal system of eigenfunctions and co-eigenfunctions but no corresponding spectral expansion.

arXiv commentThere was a gap in the proof of Theorem 4.20 in the earlier version. A different proof is now given. The relation between the eigenfunctions and Appell polynomials has been introduced in the current version