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 $\mathcal{L}={\mathcal{L}}^{\sigma ,b}+{\mathcal{L}}^{\mu }$ where $${\mathcal{L}}^{\sigma ,b}[u](x)=\text{tr}(\sigma {\sigma }^{\text{T}}{D}^{2}u(x))+b\cdot Du(x)$$ and $${\mathcal{L}}^{\mu }[u](x)=\int (u(x+z)-u-z\cdot Du(x)1_{|z|\le 1})\mathrm{d}\mu (z).$$ 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 translation invariant operators of this form that satisfy the Liouville theorem: Bounded solutions $u$ of $\mathcal{L}[u]=0$ in ${\mathbb{R}}^d$ are constant. The Liouville property is obtained as a consequence of a periodicity result that completely characterizes bounded distributional solutions of $\mathcal{L}[u]=0$ in ${\mathbb{R}}^d$. The proofs combine arguments from PDE and group theories. They are simple and short.

This is an independent and substantial update of arXiv:1807.01843. Here we treat general operators which could be both local and nonlocal, symmetric and nonsymmetric. 13 pages. v3: Update according to the suggestions of the referees. To appear in Journal de Mathématiques Pures et Appliquées