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 $L_0$ in $R^N$, as a consequence of a Liouville theorem at “t = −∞” for the corresponding Kolmogorov operators $L_0 − ∂t$ in $R^{N+1}$. In turn, this last result is proved as a corollary of a global Harnack inequality for non-negative solutions to $(L_0 − ∂t)u = 0 $ which seems to have an independent interest in its own right. We stress that our Liouville theorem for $L_0$ 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.