Harmonic functions on homogeneous spaces
Let \(G\) be a locally compact group acting on a locally compact space \(X\) and \(\sigma\) a probability measure on \(G\). A real Borel function on \(X\) is said to be \(\sigma\)-harmonic if it satisfies the convolution equation \(f=\sigma *f\). In 1963, \textit{H. Furstenberg} [Ann. Math. (2) 77, 335-386 (1963; Zbl 0192.12704)] showed that for a connected semisimple Lie group \(G\) with finite centre and maximal compact subgroup \(K\), the homogeneous space \(G/K\) has the Liouville property, that is, every bounded harmonic function on it is necessarily constant. In this paper, the authors show that Furstenberg's proof of the Liouville property for \(G/K\) is valid in a wider setting. More precisely, introducing first the concept of \(\sigma\)-admissible neighbourhood of the identity relative to \(X\), where \(X\) is a homogeneous space of \(G\), and adapting Furstenberg's arguments, they show that if \(G\) is a union of \(\sigma\)-admissible neighborhoods, then the bounded \(\sigma\)-harmonic functions on \(X\) are constant. Then they present some interesting consequences, pointing out several examples of such spaces \(X\) having the Liouville property.
scientific article; zbMATH DE number 1382945