Vol. 141, No. 1, 1990

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The Poisson flow associated with a measure

Douglas Dokken and Robert Ellis

Vol. 141 (1990), No. 1, 79–103
Abstract

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 μ on a 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(A1) for all Borel sets A, and a new non-geometric proof of a fundamental result of Furstenberg’s on semi-simple Lie groups.

Mathematical Subject Classification 2000
Primary: 22D40
Milestones
Received: 2 March 1988
Revised: 8 November 1988
Published: 1 January 1990
Authors
Douglas Dokken
Robert Ellis