Gradient estimates of harmonic functions and transition densities for Levy processes

TL;DR

This paper establishes gradient estimates for harmonic functions related to unimodal pure-jump Levy processes, using novel constructions of related processes to analyze transition densities and differential properties.

Contribution

It introduces new methods to derive gradient estimates by constructing related Levy processes and difference processes, expanding understanding of harmonic functions for jump processes.

Findings

Gradient estimates for harmonic functions under mild Levy measure conditions

Construction of an unimodal Levy process in higher dimensions with the same characteristic exponent

Development of a difference process to analyze differential properties of harmonic functions

Abstract

We prove gradient estimates for harmonic functions with respect to a $d$-dimensional unimodal pure-jump Levy process under some mild assumptions on the density of its Levy measure. These assumptions allow for a construction of an unimodal Levy process in $\R^{d+2}$ with the same characteristic exponent as the original process. The relationship between the two processes provides a fruitful source of gradient estimates of transition densities. We also construct another process called a difference process which is very useful in the analysis of differential properties of harmonic functions.