A general Choquet-Deny theorem for nilpotent groups
Let \(G\) be a locally compact second countable group and \(\mu\) a probability measure on the Borel \(\sigma\)-algebra of \(G\). A bounded Borel function \(h\) on \(G\) is said to be \(\mu\)-harmonic if \(h(g)=\int_{G}h(gx)d\mu(x)\quad\forall g\in G.\) The probability measure \(\mu\) is said to be aperiodic if the closed subgroup generated by supp(\(\mu\)) coincides with \(G\). \textit{G. Choquet} and \textit{J. Deny} [C. R. Acad. Sci., Paris 250, 799--801 (1960; Zbl 0093.12802)] proved that if \(\mu\) is aperiodic and \(G\) is Abelian, then every continuous bounded \(\mu\)-harmonic function on \(G\) is constant. \textit{E. B. Dynkin} and \textit{M. B. Malyutov} [Sov. Math., Dokl. 2, 399--402 (1961); translation from Dokl. Akad. Nauk SSSR 137, 1042--1045 (1961; Zbl 0214.44101)] extended the result to all discrete nilpotent groups with a finite number of generators. Other extensions (with conditions on \(\mu\) or \(G\)) were given by \textit{R. Azencott} [``Espaces de Poisson des groupes localement compacts.'' Berlin etc.: Springer (1970; Zbl 0208.15302)], \textit{Y. Guivarc'h} [Bull. Soc. Math. Fr. 101, 333--379 (1973; Zbl 0294.43003)] and \textit{A. Avez} [C. R. Acad. Sci., Paris, Sér. A 279, 25--28 (1974; Zbl 0292.60100)]. In the paper under review the author proves the theorem of Choquet and Deny (if \(\mu\) is aperiodic, then any continuous bounded \(\mu\)-harmonic function on \(G\) is constant) on any locally compact second countable nilpotent group, without any other assumption on \(G\) or \(\mu\). He also gives a new and very simple proof of the Choquet-Deny theorem for Abelian semigroups.
scientific article; zbMATH DE number 2119571