The Poisson flow associated with a measure (original) (raw)

This paper is devoted to the study of harmonic functions on groups. The approach is via a detailed study of the Poisson flow associated with a Borel probability measure /iona locally compact group T. Again the basic idea is that though many results associated with the study of harmonic functions on groups are couched in probabilistic terms and proved using methods of probability theory, they really belong in the domain of topological dynamics. The major results include a proof that a solvable connected Lie group admits only constants as harmonic functions for a spread out measure μ with μ(A) = μ(A~ι) for all Borel sets A, and a new non-geometric proof of a fundamental result of Furstenberg's on semi-simple Lie groups.