Excursions away from a regular point for one-dimensional symmetric Levy processes without Gaussian part

TL;DR

This paper investigates the properties of excursions of one-dimensional symmetric Lévy processes without Gaussian components, establishing key measure characteristics and path behaviors.

Contribution

It proves the extremeness of the excursion measure and demonstrates the Feller property of the harmonic transform of the killed process.

Findings

Harmonic transform of the killed process has Feller property

Excursion measure is extremal

Sample path behaviors of excursions and h-path processes analyzed

Abstract

The characteristic measure of excursions away from a regular point is studied for a class of symmetric L\'evy processes without Gaussian part. It is proved that the harmonic transform of the killed process enjoys Feller property. The result is applied to prove extremeness of the excursion measure and to prove several sample path behaviors of the excursion and the $ h $-path processes.