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

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.