The Liouville theorem and linear operators satisfying the maximum principle.

Abstract

A result by Courrège says that linear translation invariant operators satisfy the maximum principle if and only if they are of the form L = Lσ,b + Lμ where Lσ,b[u](x) = tr(σσTD2u(x)) + b · Du(x) and Lμ[u](x) = Rd\{0} u(x + z) − u(x) − z · Du(x)1|z|≤1 dμ(z). This class of operators coincides with the infinitesimal generators of Lévy processes in probability theory. In this paper we give a complete characterization of the operators of this form that satisfy the Liouville theorem: Bounded solutions u of L[u] = 0 in Rd are constant. The Liouville property is obtained as a consequence of a periodicity result that completely characterizes bounded distributional solutions of L[u] = 0 in Rd. The proofs combine arguments from PDEs and group theory. They are simple and short.