Harnack Inequalities and Applications for Ornstein-Uhlenbeck Semigroups with Jump

TL;DR

This paper improves and generalizes Harnack inequalities for Ornstein-Uhlenbeck semigroups with jumps, establishing various functional inequalities and properties, including applications to Gaussian cases and perturbations with linear growth.

Contribution

It extends existing Harnack inequalities to more general jump processes and explores their implications for semigroup properties and Gaussian cases.

Findings

Improved Harnack inequality for jump-type Ornstein-Uhlenbeck semigroups.

Established log-Harnack, strong Feller, and heat kernel inequalities.

Proved HWI inequality for Gaussian case.

Abstract

The Harnack inequality established in [13] for generalized Mehler semigroup is improved and generalized. As applications, the log-Harnack inequality, the strong Feller property, the hyper-bounded property, and some heat kernel inequalities are presented for a class of O-U type semigroups with jump. These inequalities and semigroup properties are indeed equivalent, and thus sharp, for the Gaussian case. As an application of the log-Harnack inequality, the HWI inequality is established for the Gaussian case. Perturbations with linear growth are also investigated.