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Harmonic measures versus quasiconformal measures for hyperbolic groups. (Mesures harmoniques et mesures quasiconformes sur les groupes hyperboliques.) (English. French summary) Zbl 1243.60005
There are two main classes of measures on the Gromov boundary of a non-elementary hyperbolic group: quasiconformal measures and harmonic measures. An important class of quasiconformal measures is the one of Patterson-Sullivan measures. Harmonic measures are induced by random walks on the group. The object of the paper under review is to study these two classes of measures and the interplay between them. The authors show that the Hausdorff dimension of the harmonic measure induced by a finitely supported and symmetric random walk on the group satisfies a “dimension-entropy-rate of escape” formula and they characterize those harmonic measures of maximal dimension. Their approach combines probabilistic and geometric techniques. In particular, the authors obtain a sharp control of the deviation of simple paths of the random walk with respect to geodesics. The approach uses a metric on the group, called the Green metric, which provides a geometric framework for transient random walks. This metric was introduced in the paper [S. Blachère and S. Brofferio, Probab. Theory Relat. Fields 137, No. 3-4, 323–343 (2007; Zbl 1106.60078)]. The metric is non-degenerate in the case of random walks. It allows the authors to interpret harmonic measures as quasiconformal measures (actually, Patterson-Sullivan measures) on the boundary of the group.

MSC:
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
20F67 Hyperbolic groups and nonpositively curved groups
31C12 Potential theory on Riemannian manifolds and other spaces
31C45 Other generalizations (nonlinear potential theory, etc.)
60J50 Boundary theory for Markov processes
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