Spectral theory of non-local Ornstein-Uhlenbeck operators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911585418608640 |
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| author | Sarkar, Rohan |
| author_facet | Sarkar, Rohan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_15183 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral theory of non-local Ornstein-Uhlenbeck operators Sarkar, Rohan Probability Analysis of PDEs Functional Analysis Spectral Theory 35P05 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. |
| title | Spectral theory of non-local Ornstein-Uhlenbeck operators |
| topic | Probability Analysis of PDEs Functional Analysis Spectral Theory 35P05 |
| url | https://arxiv.org/abs/2502.15183 |