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

TL;DR

This paper investigates the Cauchy problem for non-local Ornstein-Uhlenbeck operators, establishing classical solvability without finite moment conditions and characterizing the process's laws via Fokker-Planck equations.

Contribution

It proves classical solvability of the problem without requiring the Lévy measure to have a finite first moment and identifies an invariant core of regular functions for the semigroup.

Findings

Classical solvability achieved without finite first moment condition.

Identified an invariant core of regular functions for the transition semigroup.

Characterized marginal laws via Fokker-Planck-Kolmogorov equations.

Abstract

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.