Green function for gradient perturbation of unimodal L\'{e}vy processes

TL;DR

This paper establishes the comparability of Green functions for isotropic unimodal Lévy processes and their gradient perturbations within bounded smooth domains, under certain conditions on the drift function.

Contribution

It introduces conditions under which the Green functions of Lévy processes and their gradient perturbations are comparable, extending understanding of their behavior in bounded smooth domains.

Findings

Green functions are comparable under specified conditions.

Gradient perturbations do not significantly alter Green functions.

Results apply to processes with weak lower scaling order greater than one.

Abstract

We prove that the Green function of a generator of isotropic unimodal L\'{e}vy processes with the weak lower scaling order bigger than one and the Green function of its gradient perturbations are comparable for bounded smooth open sets if the drift function is from an appropriate Kato class.