Duke Mathematical Journal

Harmonicity of Gibbs measures

Chris Connell and Roman Muchnik

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We show that any continuous measure ν in the class of a generalized Gibbs stream on the boundary of a CAT(κ) group G arises as a harmonic measure for a random walk on G. Under an additional mild hypothesis on G and for ν, Hölder equivalent to a Gibbs measure, we show that (G,ν) arises as a Poisson boundary for a random walk on G. We also prove a new approximation theorem for general metric measure spaces giving quite flexible conditions for a set of functions to be a positive basis for the cone of positive continuous functions

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Duke Math. J., Volume 137, Number 3 (2007), 461-509.

First available in Project Euclid: 6 April 2007

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Primary: 60J50: Boundary theory 20F67: Hyperbolic groups and nonpositively curved groups 37A35: Entropy and other invariants, isomorphism, classification 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)


Connell, Chris; Muchnik, Roman. Harmonicity of Gibbs measures. Duke Math. J. 137 (2007), no. 3, 461--509. doi:10.1215/S0012-7094-07-13732-3. https://projecteuclid.org/euclid.dmj/1175865518

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