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arXiv · 2504.13673

One-side Liouville Theorem for hypoelliptic Ornstein--Uhlenbeck operators having drifts with imaginary spectrum

Alessia E. Kogoj·Ermanno Lanconelli·Giulio Tralli
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Abstract

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.

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Alessia E. Kogoj, Ermanno Lanconelli, Giulio Tralli. 2025-04-18. One-side Liouville Theorem for hypoelliptic Ornstein--Uhlenbeck operators having drifts with imaginary spectrum. https://arxiv.org/abs/2504.13673

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