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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2502.15183 |
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- We consider non-local Ornstein-Uhlenbeck (OU) operators that correspond to Ornstein-Uhlenbeck processes driven by Lévy processes. These are ergodic Markov processes and the OU operator is in general non-normal in the $L^2$ space weighted with the invariant distribution. Under some mild assumptions on the Lévy process, we carry out in-depth analysis of the spectrum, spectral multilicities, eigenfunctions and co-eigenfunctions (eigenfunctions of the adjoint), and the existence of spectral expansion of the semigroups. When the drift matrix $B$ is diagonalizable, we derive explicit formulas for eigenfunctions and co-eigenfunctions which are also biorthogonal, and such results continue to hold when the Lévy process is a pure jump process. A key ingredient in our approach is \emph{intertwining relationship}: we prove that every Lévy-OU semigroup is intertwined with a diffusion OU semigroup. Additionally, we study the compactness properties of these semigroups and provide some necessary and sufficient conditions for compactness.