The Liouville theorem and linear operators satisfying the maximum principle

A result by Courrege says that linear translation invariant operators satisfy the maximum principle if and only if they are of the form L = L-sigma,L-b + L-mu where L-sigma,L-b[u](x) = tr(sigma sigma(T) D(2)u(x)) + b . Du(x) and L-mu[u](x) = integral(Rd\{0}) (u(x + z) - u(x) - z . Du(x)1(vertical bar <=vertical bar 1)) d(mu)(z). This class of operators coincides with the infinitesimal generators of Levy 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 R-d 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 R-d. The proofs combine arguments from PDEs and group theory. They are simple and short. (C) 2020 The Authors. Published by Elsevier Masson SAS.