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The Liouville theorem and linear operators satisfying the maximum principle

A result by Courrège says that linear translation invariant operators satisfy the maximum principle if and only if they are of the form $\mathcal{L}=\mathcal{L}^{σ,b}+\mathcal{L}^μ$ where $$ \mathcal{L}^{σ,b}[u](x)=\text{tr}(σσ^{\texttt{T}} D^2u(x))+b\cdot Du(x) $$ and $$ \mathcal{L}^μ[u](x)=\int \big(u(x+z)-u-z\cdot Du(x) \mathbf{1}_{|z| \leq 1}\big) \,\mathrm{d} μ(z). $$ 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.