Green function for gradient perturbation of unimodal L\'evy processes in the real line

TL;DR

This paper establishes the comparability of Green functions for symmetric unimodal Lévy processes and their gradient perturbations on bounded smooth subsets of the real line, under certain conditions on the drift function.

Contribution

It provides a new comparison result for Green functions of Lévy processes and their gradient perturbations with specific drift conditions.

Findings

Green functions are comparable under given conditions.

Results apply to bounded $C^{1,1}$ subsets of the real line.

Conditions on the drift function are specified within a Kato class.

Abstract

We prove that the Green function of a generator of symmetric unimodal L\'evy processes with the weak lower scaling order bigger than one and the Green function of its gradient perturbations are comparable for bounded $C^{1,1}$ subsets of the real line if the drift function is from an appropriate Kato class.