On the Liouville property for non-local L\'evy generators

TL;DR

This paper establishes a precise criterion for when the generator of a Lévy process exhibits the Liouville property, linking it to the zeros of the characteristic exponent, thus advancing understanding of harmonic functions for these processes.

Contribution

It provides a necessary and sufficient condition for the Liouville property of Lévy generators, extending previous criteria and characterizing zeros of the characteristic exponent.

Findings

Established a criterion for the Liouville property of Lévy generators.

Characterized the zeros of the characteristic exponent of Lévy processes.

Connected the Liouville property with the structure of the process's characteristic exponent.

Abstract

We prove a necessary and sufficient condition for the Liouville property of the infinitesimal generator of a L\'evy process and subordinate L\'evy processes. Combining our criterion with the necessary and sufficient condition obtained by Alibaud et al., we obtain a characterization of (orthogonal subgroup of the) the set of zeros of the characteristic exponent of the L\'evy process.