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The Liouville theorem and linear operators satisfying the maximum principle
Journal de Mathématiques Pures et Appliquées ( IF 2.1 ) Pub Date : 2020-08-24 , DOI: 10.1016/j.matpur.2020.08.008
Nathaël Alibaud , Félix del Teso , Jørgen Endal , Espen R. Jakobsen

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 whereLσ,b[u](x)=tr(σσTD2u(x))+bDu(x) andLμ[u](x)=Rd{0}(u(x+z)u(x)zDu(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.



中文翻译:

满足最大原理的Liouville定理和线性算子

Courrège的结果表明,线性平移不变算符在且仅当其形式为时才满足最大原理 大号=大号σb+大号μ 哪里大号σb[ü]X=TRσσŤd2üX+bdüX大号μ[ü]X=[Rd{0}üX+ž-üX-ždüX1个|ž|1个dμž这类算子与概率论中Lévy过程的无穷小生成器一致。在本文中,我们给出了满足刘维定理这种形式的运营商一个完整的表征:界解ü大号[ü]=0[Rd是恒定的。Liouville属性是作为周期结果的结果获得的,该结果完全刻画了大号[ü]=0[Rd。证明结合了PDE和组理论的论点。它们简单而简短。

更新日期:2020-08-24
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