On harmonic function for the killed process upon hitting zero of asymmetric L\'evy processes

TL;DR

This paper proves that for a specific class of asymmetric Lévy processes, the renormalized zero resolvent function is harmonic for the process killed upon hitting zero, enhancing understanding of potential theory in stochastic processes.

Contribution

It establishes the harmonicity of the renormalized zero resolvent for asymmetric Lévy processes with regular origin, a novel result in the theory of Lévy processes.

Findings

Renormalized zero resolvent is harmonic for the killed process.

Applicable to a class of asymmetric Lévy processes.

Provides new insights into potential theory for Lévy processes.

Abstract

For a certain class of asymmetric L\'evy processes where the origin is regular for itself, the renormalized zero resolvent is proved to be harmonic for the killed process upon hitting zero.