On the Liouville property for non-local Lévy generators

Abstract

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