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.