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 quasiconformal measures on the boundary of the group.
On établit une formule de la dimension de la mesure harmonique d'une marche aléatoire de loi de support fini et symétrique sur un groupe hyperbolique. On caractérise aussi les lois pour lesquelles la dimension est maximale. Notre approche repose sur la distance de Green, une distance qui permet de développer un point de vue géométrique sur les marches aléatoires et, en particulier, d'interpréter les mesures harmoniques comme des mesures quasiconformes.
Keywords: hyperbolic groups, random walks on groups, harmonic measures, quasiconformal measures, dimension of a measure, Martin boundary, brownian motion, Green metric
Mot clés : groupes hyperboliques, marches aléatoires sur les groupes, mesures harmoniques, mesures quasiconformes, dimension d'une mesure, bord de Martin, mouvement brownien, distance de Green
@article{ASENS_2011_4_44_4_683_0, author = {Blach\`ere, S\'ebastien and Ha{\"\i}ssinsky, Peter and Mathieu, Pierre}, title = {Harmonic measures versus quasiconformal measures for hyperbolic groups}, journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure}, pages = {683--721}, publisher = {Soci\'et\'e math\'ematique de France}, volume = {Ser. 4, 44}, number = {4}, year = {2011}, doi = {10.24033/asens.2153}, mrnumber = {2919980}, zbl = {1243.60005}, language = {en}, url = {http://www.numdam.org/articles/10.24033/asens.2153/} }
TY - JOUR AU - Blachère, Sébastien AU - Haïssinsky, Peter AU - Mathieu, Pierre TI - Harmonic measures versus quasiconformal measures for hyperbolic groups JO - Annales scientifiques de l'École Normale Supérieure PY - 2011 SP - 683 EP - 721 VL - 44 IS - 4 PB - Société mathématique de France UR - http://www.numdam.org/articles/10.24033/asens.2153/ DO - 10.24033/asens.2153 LA - en ID - ASENS_2011_4_44_4_683_0 ER -
%0 Journal Article %A Blachère, Sébastien %A Haïssinsky, Peter %A Mathieu, Pierre %T Harmonic measures versus quasiconformal measures for hyperbolic groups %J Annales scientifiques de l'École Normale Supérieure %D 2011 %P 683-721 %V 44 %N 4 %I Société mathématique de France %U http://www.numdam.org/articles/10.24033/asens.2153/ %R 10.24033/asens.2153 %G en %F ASENS_2011_4_44_4_683_0
Blachère, Sébastien; Haïssinsky, Peter; Mathieu, Pierre. Harmonic measures versus quasiconformal measures for hyperbolic groups. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 44 (2011) no. 4, pp. 683-721. doi : 10.24033/asens.2153. http://www.numdam.org/articles/10.24033/asens.2153/
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