Harmonic measures versus quasiconformal measures for hyperbolic groups
[Mesures harmoniques et mesures quasiconformes sur les groupes hyperboliques]
Blachère, Sébastien ; Haïssinsky, Peter ; Mathieu, Pierre
Annales scientifiques de l'École Normale Supérieure, Tome 44 (2011), p. 683-721 / Harvested from Numdam
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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.

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.

Publié le : 2011-01-01
DOI : https://doi.org/10.24033/asens.2153
Classification:  20F67,  60B15,  11K55,  20F69,  28A75,  60J50,  60J65
Mots clés: groupes hyperboliques, marches aléatoires sur les groupes, mesures harmoniques, mesures quasiconformes, dimension d'une mesure, bord de Martin, mouvement brownien, distance de Green 
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     title = {Harmonic measures versus quasiconformal measures for hyperbolic groups},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {44},
     year = {2011},
     pages = {683-721},
     doi = {10.24033/asens.2153},
     mrnumber = {2919980},
     zbl = {1243.60005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ASENS_2011_4_44_4_683_0}
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Blachère, Sébastien; Haïssinsky, Peter; Mathieu, Pierre. Harmonic measures versus quasiconformal measures for hyperbolic groups. Annales scientifiques de l'École Normale Supérieure, Tome 44 (2011) pp. 683-721. doi : 10.24033/asens.2153. http://gdmltest.u-ga.fr/item/ASENS_2011_4_44_4_683_0/

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