On a long standing conjecture: positive Liouville Theorem for hypoelliptic Ornstein-Uhlenbeck operators
Abstract
Let $\mathcal{L}$ be the hypoelliptic Ornstein-Uhlenbeck operator associated with the pair of matrices (A,B).In 2004, Priola and Zabczyk proved the following Liouville-type theorem: every bounded entire solution of $\mathcal{L}u=0$ is constant if and only if (*) every eigenvalue of B has real part less than or equal to zero.This remarkable result raised the following problem, which is still not completely solved: if condition (*) holds, is it true that every non-negative entire solution of $\mathcal{L}u=0$ is constant?In this note, along with a review of the current state of research on this problem, we present some recent new results.
Fields of Study & Subjects
Nonlinear Partial Differential Equations
Geometric Analysis and Curvature Flows
Spectral Theory in Mathematical Physics
Publication details
- Journal / source
- Bruno Pini mathematical analysis Seminar
- Volume
- 16
- Issue
- 1
- Pages
- 136–147
- Publication date
- 25 February 2026
- Language
- EN