  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.
