Harnack inequality and Liouville-type theorems for Ornstein-Uhlenbeck and Kolmogorov operators

Abstract

We prove, with a purely analytic technique, a one-side Liouville theorem for a class of Ornstein--Uhlenbeck operators ${\mathcal{L}}_0$ in ${\mathbb{R}}^N$, as a consequence of a Liouville theorem at "$t=-\infty$" for the corresponding Kolmogorov operators ${\mathcal{L}}_0-{\partial }_t$ in ${\mathbb{R}}^{N+1}$. In turn, this last result is proved as a corollary of a global Harnack inequality for non-negative solutions to $({\mathcal{L}}_0-{\partial }_t)u=0$ which seems to have an independent interest in its own right. We stress that our Liouville theorem for ${\mathcal{L}}_0$ cannot be obtained by a probabilistic approach based on recurrence if $N>2$. We provide a self-contained proof of a Liouville theorem involving recurrent Ornstein--Uhlenbeck stochastic processes in the Appendix.