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

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.