Harmonicity of Gibbs measures. (English) Zbl 1133.60032
This paper is concerned with the question whether a given probability measure on a geometric boundary (of a group) arises as a measure defining a Poisson boundary, or at least arises as a harmonic measure. It extends earlier work of the authors. Let \(H\) be a \(\text{CAT}(-1)\) space, \(G\) be a group of isometrics acting cocompactly on \(H\) and \(\Phi: SH\to\mathbb{R}\) be a certain Hölder continuous function. Let \(\nu^\Phi\) denote the Gibbs measure for the geodesic flow. By disintegration, \(\nu^\Phi\) gives rise to a family of measures \(\nu^\Phi_p\), \(p\in H\), on the boundary \(\partial H\) of \(H\). The main theorem states that for any \(p\in H\) there exists a measure \(m_p\) on \(G\) such that \(\nu^\Phi_p\) is \(m_p\)-stationary and \((\partial H,\nu^\Phi_p)\) is the Poisson boundary of the random walk \((G,m_p)\).
A slightly more general version in terms of harmonic convolutions is shown as well. The major part of the paper is used to prove the results.
A slightly more general version in terms of harmonic convolutions is shown as well. The major part of the paper is used to prove the results.
Reviewer: Manfred Denker (Göttingen)
MSC:
| 60J50 | Boundary theory for Markov processes |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
| 37A35 | Entropy and other invariants, isomorphism, classification in ergodic theory |
| 41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
Keywords:
Poisson boundary; random walk; Gibbs measure; harmonic measure; geometric boundary; hyperbolic groupReferences:
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