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 L0 in ℝN, as a consequence of a Liouville theorem at “t=−∞” for the corresponding Kolmogorov operators L0−∂t in ℝN+1. In turn, this last result is proved as a corollary of a global Harnack inequality for non-negative solutions to (L0−∂t)u=0 which seems to have an independent interest in its own right. We stress that our Liouville theorem for L0 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