Harmonicity of Gibbs measures (Q881923)

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.