On the Cauchy problem for non-local Ornstein--Uhlenbeck operators

We study the Cauchy problem involving non-local Ornstein–Uhlenbeck operators in finite and infinite dimensions. We prove classical solvability without requiring that the Lévy 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.