We study harmonic functions and Poisson boundaries for Borel probability measures on general (i.e., not necessarily locally compact) topological groups, and we prove that a second-countable topological group is amenable if and only if it admits a fully supported, regular Borel probability measure with trivial Poisson boundary. This generalizes work of Kaimanovich--Vershik and Rosenblatt, confirms a general topological version of Furstenberg's conjecture, and entails a characterization of the amenability of isometry groups in terms of the Liouville property for induced actions. Moreover, our result has non-trivial consequences concerning Liouville actions of discrete groups on countable sets

中文翻译：

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拓扑群上的刘维尔性质和随机游走

我们研究了一般（即，不一定是局部紧凑的）拓扑群的 Borel 概率测度的调和函数和泊松边界，我们证明了第二可数拓扑群是适用的，当且仅当它承认一个完全支持的正则 Borel 概率测度具有平凡的泊松边界。这概括了 Kaimanovich--Vershik 和 Rosenblatt 的工作，证实了 Furstenberg 猜想的一般拓扑版本，并且需要根据诱导作用的 Liouville 性质对等距群的适应性进行表征。此外，我们的结果对可数集上离散群的刘维尔行为具有非平凡的后果