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

Enrico Priola, Stefano Tracà

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 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.

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

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 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.

Paper Structure

This paper contains 10 sections, 14 theorems, 135 equations.

Table of Contents

  1. Introduction and notation
  2. Preliminaries
  3. Ornstein-Uhlenbeck operators in $\mathbb{R}^d$
  4. Classical solvability of the Cauchy problem
  5. An invariant core in $C_0(\mathbb{R}^d)$
  6. Infinite dimensional Ornstein-Uhlenbeck processes
  7. The Cauchy problem in infinite dimensions
  8. The Ornstein-Uhlenbeck generator ${\cal L}$ in $C_b(H)$
  9. A $\pi$-core for the generator $\cal L$
  10. Kolmogorov equations for measures

Key Result

Theorem \oldthetheorem

Let $f\in C_b^2(\mathbb{R}^d)$. Then, for any $x \in \mathbb{R}^d$, the mapping: $t \mapsto \mathcal{L}_0 (P_t f ) (x)$ is continuous on $[0, + \infty)$ and $\lim_{t \to 0^+} \mathcal{L}_0 (P_t f ) (x) = \mathcal{L}_0 f (x)$. Moreover,

Theorems & Definitions (31)

  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof : Proof of Theorem \ref{['PCAUCHY']}
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • ...and 21 more