Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal L\'evy processes

TL;DR

This paper provides sharp estimates for hitting probabilities, potential kernels, and harmonic functions of isotropic unimodal Lévy processes, extending boundary Harnack inequalities and Green function estimates in half-spaces.

Contribution

It introduces new sharp bounds for potential kernels and Green functions under mild regularity conditions, and applies boundary Harnack inequalities to characterize harmonic functions.

Findings

Sharp two-sided estimates of hitting probabilities and Green functions.

Regularity results for harmonic functions with respect to Lévy processes.

Application of boundary Harnack inequality to half-space Green function estimates.

Abstract

In the first part of this article, we prove two-sided estimates of hitting probabilities of balls, the potential kernel and the Green function for a ball for general isotropic unimodal L\'evy processes. Our bounds are sharp under the absence of the Gaussian component and a mild regularity condition on the density of the L\'{e}vy measure: its radial profile needs to satisfy a scaling-type condition, which is equivalent to $O$-regular variation at zero and at infinity with lower indices greater than $-d - 2$. We also prove a supremum estimate and a regularity result for functions harmonic with respect to a general isotropic unimodal L\'evy process. In the second part we apply the recent results on the boundary Harnack inequality and Martin representation of harmonic functions for the class of isotropic unimodal L\'evy processes characterised by a localised version of the scaling-type condition mentioned above. As a sample application, we provide sharp two-sided estimates of the Green function of a half-space. Our results are expressed in terms of Pruitt's functions $K(r)$ and $L(r)$, measuring local activity and the amount of large jumps of the L\'evy process, respectively.