  We study the Cauchy problem involving non-local Ornstein-Uhlenbeck operators
in finite and infinite dimensions. We prove classical solvability without
requiring that the L\'evy measure corresponding to the large jumps part has a
first finite moment. Moreover, we determine a core of regular functions which
is invariant for the associated transition Markov semigroup. Such a core allows
to characterize the marginal laws of the Ornstein-Uhlenbeck stochastic process
as unique solutions to Fokker-Planck-Kolmogorov equations for measures.
