Harmonic measures versus quasiconformal measures for hyperbolic groups

Sébastien Blachère (1, 2) , Peter Haïssinsky (3) , Pierre Mathieu (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 EURANDOM - European Institute for Statistics, Probability, Stochastic Operations Research and its Applications
3 Picard

We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on random walks and, in particular, which allows us to interpret harmonic measures as \qc measures on the boundary of the group.

Mots clés: Hyperbolic groups; random walks on groups; harmonic measures; quasiconformal measures; dimension of a measure; Martin boundary; Brownian motion; Green metric

Domaines: Probabilités [math.PR]; Géométrie métrique [math.MG]; Théorie des groupes [math.GR]

Sébastien Blachère, Peter Haïssinsky, Pierre Mathieu. Harmonic measures versus quasiconformal measures for hyperbolic groups. 2008. ⟨hal-00290127v1⟩