Harnack inequalities for Ornstein–Uhlenbeck processes driven by Lévy processes

Abstract

By using the existing sharp estimates of the density function for rotationally invariant symmetric α -stable Lévy processes and rotationally invariant symmetric truncated α -stable Lévy processes, we obtain that the Harnack inequalities hold for rotationally invariant symmetric α -stable Lévy processes with α ∈ ( 0 , 2 ) and Ornstein–Uhlenbeck processes driven by rotationally invariant symmetric α -stable Lévy process, while the logarithmic Harnack inequalities are satisfied for rotationally invariant symmetric truncated α -stable Lévy processes.