## Duke Mathematical Journal

### Harmonicity of Gibbs measures

#### Abstract

We show that any continuous measure $\nu$ in the class of a generalized Gibbs stream on the boundary of a CAT($-\kappa$) group $G$ arises as a harmonic measure for a random walk on $G$. Under an additional mild hypothesis on $G$ and for $\nu$, Hölder equivalent to a Gibbs measure, we show that $(\partial G,\nu)$ 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

#### Article information

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

Dates
First available in Project Euclid: 6 April 2007

https://projecteuclid.org/euclid.dmj/1175865518

Digital Object Identifier
doi:10.1215/S0012-7094-07-13732-3

Mathematical Reviews number (MathSciNet)
MR2309151

Zentralblatt MATH identifier
1133.60032

#### Citation

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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