Harnack Inequalities for Ornstein-Uhlenbeck Processes Driven by L\'{e}vy Processes

TL;DR

This paper establishes Harnack inequalities for Ornstein-Uhlenbeck processes driven by symmetric α-stable Lévy processes, extending the understanding of their regularity properties using sharp density estimates.

Contribution

It proves Harnack inequalities for Ornstein-Uhlenbeck processes driven by symmetric α-stable Lévy processes, utilizing existing density estimates for these processes.

Findings

Harnack inequalities hold for symmetric α-stable Lévy driven Ornstein-Uhlenbeck processes.

Logarithmic Harnack inequalities are valid for truncated α-stable Lévy processes.

Results extend regularity theory for Lévy-driven stochastic processes.

Abstract

By using the existing sharp estimates of density function for rotationally invariant symmetric $\alpha$-stable L\'{e}vy processes and rotationally invariant symmetric truncated $\alpha$-stable L\'{e}vy processes, we obtain that Harnack inequalities hold for rotationally invariant symmetric $\alpha$-stable L\'{e}vy processes with $\alpha\in(0,2)$ and Ornstein-Uhlenbeck processes driven by rotationally invariant symmetric $\alpha$-stable L\'{e}vy process, while logarithmic Harnack inequalities are satisfied for rotationally invariant symmetric truncated $\alpha$-stable L\'{e}vy processes.