Potential theory of one-dimensional geometric stable processes

TL;DR

This paper derives optimal estimates for the Green function and Poisson kernel of one-dimensional geometric stable processes, establishing key inequalities and principles relevant for potential theory.

Contribution

It provides the first comprehensive estimates for Green functions and Poisson kernels of geometric stable processes, including the proof of scale-invariant Harnack inequality and boundary Harnack principle.

Findings

Optimal Green function estimates for geometric stable processes

Poisson kernel estimates for intervals and half-lines

Proof of scale-invariant Harnack inequality and boundary Harnack principle

Abstract

The purpose of this paper is to find optimal estimates for the Green function and the Poisson kernel for a half-line and intervals of the geometric stable process with parameter $\alpha\in(0,2]$. This process has an infinitesimal generator of the form $-\log(1+(-\Delta)^{\alpha/2})$. As an application we prove the scale invariant Harnack inequality as well as the boundary Harnack principle.