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 = -infinity" for the corresponding Kolmogorov operators L-0 partial derivative(t) in RN+1. In turn, this last result is proved as a corollary of a global Harnack inequality for non-negative solutions to (L-0 partial derivative(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.