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

2025/04/18 by Kogoj, Alessia E., Lanconelli, Ermanno, Tralli, Giulio
#35B53 #35H10 #35K99 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2504.13673

Abstract

We prove the Liouville theorem for 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 bounded solutions. Our approach is based on a Liouville property at ``t=-∞" 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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