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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.13673 |
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| _version_ | 1866917989976113152 |
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| author | Kogoj, Alessia E. Lanconelli, Ermanno Tralli, Giulio |
| author_facet | Kogoj, Alessia E. Lanconelli, Ermanno Tralli, Giulio |
| contents | We prove the Liouville theorem for \emph{non-negative} solutions to (possibly degenerate) Ornstein-Uhlenbeck equations whose linear drift has imaginary spectrum. This provides an answer to a question raised by Priola and Zabczyk since the proof of their Theorem characterizing the Ornstein-Uhlenbeck operators having the Liouville property for \emph{bounded} solutions. Our approach is based on a Liouville property at ``$t=-\infty$" for the solutions to the relevant Kolmogorov equation which, in turn, derives from a new parabolic Harnack-type inequality for its non-negative ancient solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13673 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | One-side Liouville Theorem for hypoelliptic Ornstein--Uhlenbeck operators having drifts with imaginary spectrum Kogoj, Alessia E. Lanconelli, Ermanno Tralli, Giulio Analysis of PDEs Probability 35B53, 35H10, 35K99 We prove the Liouville theorem for \emph{non-negative} solutions to (possibly degenerate) Ornstein-Uhlenbeck equations whose linear drift has imaginary spectrum. This provides an answer to a question raised by Priola and Zabczyk since the proof of their Theorem characterizing the Ornstein-Uhlenbeck operators having the Liouville property for \emph{bounded} solutions. Our approach is based on a Liouville property at ``$t=-\infty$" for the solutions to the relevant Kolmogorov equation which, in turn, derives from a new parabolic Harnack-type inequality for its non-negative ancient solutions. |
| title | One-side Liouville Theorem for hypoelliptic Ornstein--Uhlenbeck operators having drifts with imaginary spectrum |
| topic | Analysis of PDEs Probability 35B53, 35H10, 35K99 |
| url | https://arxiv.org/abs/2504.13673 |