Amenability and superharmonic functions
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Authors
Northshield, Sam
Issue Date
1993
Type
Article
Language
en_US
Keywords
Amenable group , superharmonic function , Martin boundary , random walk
Alternative Title
Abstract
Let G be a countable group and u a symmetric and aperiodic probability measure on G . We show that G is amenable if and only if every positive superharmonic function is nearly constant on certain arbitrarily large subsets of G. We use this to show that if G is amenable, then the Martin boundary of G contains a fixed point. More generally, we show that G is amenable if and only if each member of a certain family of G-spaces contains a fixed point.
Description
This article has been published in the October 1993 issue of Proceedings of the American Mathematical Society.
Citation
Northshield, S. (1993). Amenability and superharmonic functions. Proceedings of the American Mathematical Society, 119(2). http://doi.org/10.1090/S0002-9939-1993-1164149-7
Publisher
Proceedings of the American Mathematical Society