The (strong) Liouville property for a class of non-local operators

TL;DR

This paper establishes a precise criterion for when certain non-local operators, associated with Lévy processes, exhibit the Liouville and strong Liouville properties, linking these to the zeros of their characteristic exponents.

Contribution

It provides a necessary and sufficient condition for the Liouville properties of generators of Lévy and subordinate Lévy processes, extending previous characterizations.

Findings

Characterization of zeros of the characteristic exponent

Necessary and sufficient condition for Liouville properties

Extension to subordinate Lévy processes

Abstract

We prove a necessary and sufficient condition for the Liouville and strong Liouville properties 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 set of zeros of the characteristic exponent of the L\'evy process.