One-side Liouville Theorem for hypoelliptic Ornstein--Uhlenbeck operators having drifts with imaginary spectrum
Alessia E. Kogoj, Ermanno Lanconelli, Giulio Tralli
TL;DR
The paper addresses whether hypoelliptic Ornstein–Uhlenbeck operators with purely imaginary drift spectrum satisfy a one-sided Liouville property for nonnegative solutions. It develops a two-tier strategy: (i) establish a Liouville result for the Kolmogorov-type operator $\mathcal{K}=\mathcal{L}-\partial_t$ at $t=-\infty$ by proving a parabolic Harnack inequality for nonnegative ancient solutions, and (ii) transfer this to the original operator $\mathcal{L}$ by exploiting large-time matrix-function estimates and a robust mean-value framework. Key contributions include uniform large-time bounds for the Kalman-type matrices, a global determinant-doubling property, precise kernel bounds across parabolic scales, and geometric control of intrinsic paraboloids via onion-like lemmas. Collectively, these results extend Liouville-type phenomena to the one-sided setting under imaginary spectrum constraints, enriching the theory of hypoelliptic diffusions and Kolmogorov-type equations with nontrivial degeneracy.
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.