A Liouville theorem for Lévy generators

Abstract

Under mild assumptions, we establish a Liouville theorem for the "Laplace" equation $Au=0$ associated with the infinitesimal generator $A$ of a Lévy process: If $u$ is a weak solution to $Au=0$ which is at most of (suitable) polynomial growth, then $u$ is a polynomial. As a by-product, we obtain new regularity estimates for semigroups associated with Lévy processes.

Proof of Proposition 2.2 has been substantially simplified

References in corpus (4)

• Strong convergence of the Euler--Maruyama approximation for a class of Lévy-driven SDEs
• Schauder estimates for equations associated with Lévy generators
• A probabilistic proof of Schoenberg's theorem
• On the Liouville property for non-local Lévy generators

Cited by in corpus (1)

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