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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2301.08540 |
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| _version_ | 1866929606022397952 |
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| author | Grzywny, Tomasz Kwaśnicki, Mateusz |
| author_facet | Grzywny, Tomasz Kwaśnicki, Mateusz |
| contents | Let $L$ be a Lévy operator. A function $h$ is said to be harmonic with respect to $L$ if $L h = 0$ in an appropriate sense. We prove Liouville's theorem for positive functions harmonic with respect to a general Lévy operator $L$: such functions are necessarily mixtures of exponentials. For signed harmonic functions we provide a fairly general result, which encompasses and extends all Liouville-type theorems previously known in this context, and which allows to trade regularity assumptions on $L$ for growth restrictions on $h$. Finally, we construct an explicit counterexample which shows that Liouville's theorem for signed functions harmonic with respect to a general Lévy operator $L$ does not hold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_08540 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Liouville's theorems for Lévy operators Grzywny, Tomasz Kwaśnicki, Mateusz Analysis of PDEs Probability Let $L$ be a Lévy operator. A function $h$ is said to be harmonic with respect to $L$ if $L h = 0$ in an appropriate sense. We prove Liouville's theorem for positive functions harmonic with respect to a general Lévy operator $L$: such functions are necessarily mixtures of exponentials. For signed harmonic functions we provide a fairly general result, which encompasses and extends all Liouville-type theorems previously known in this context, and which allows to trade regularity assumptions on $L$ for growth restrictions on $h$. Finally, we construct an explicit counterexample which shows that Liouville's theorem for signed functions harmonic with respect to a general Lévy operator $L$ does not hold. |
| title | Liouville's theorems for Lévy operators |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2301.08540 |