Harmonic measures versus quasiconformal measures for hyperbolic groups
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 44 (2011) no. 4, p. 683-721

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.

DOI : https://doi.org/10.24033/asens.2153
Classification:  20F67,  60B15,  11K55,  20F69,  28A75,  60J50,  60J65
Keywords: hyperbolic groups, random walks on groups, harmonic measures, quasiconformal measures, dimension of a measure, Martin boundary, brownian motion, Green metric
@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},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 44},
     number = {4},
     year = {2011},
     pages = {683-721},
     doi = {10.24033/asens.2153},
     zbl = {1243.60005},
     mrnumber = {2919980},
     language = {en},
     url = {http://www.numdam.org/item/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/item/ASENS_2011_4_44_4_683_0/

[1] A. Ancona, Positive harmonic functions and hyperbolicity, in Potential theory-surveys and problems (Prague, 1987), Lecture Notes in Math. 1344, Springer, 1988, 1-23. | MR 973878 | Zbl 0677.31006

[2] A. Ancona, Théorie du potentiel sur les graphes et les variétés, in École d'été de Probabilités de Saint-Flour XVIII-1988, Lecture Notes in Math. 1427, Springer, 1990, 1-112. | MR 1100282 | Zbl 0719.60074

[3] M. T. Anderson & R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math. 121 (1985), 429-461. | MR 794369 | Zbl 0587.53045

[4] W. Ballmann, On the Dirichlet problem at infinity for manifolds of nonpositive curvature, Forum Math. 1 (1989), 201-213. | MR 990144 | Zbl 0661.53026

[5] W. Ballmann & F. Ledrappier, Discretization of positive harmonic functions on Riemannian manifolds and Martin boundary, in Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), Sémin. Congr. 1, Soc. Math. France, 1996, 77-92. | MR 1427756 | Zbl 0885.53037

[6] G. Besson, G. Courtois & S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal. 5 (1995), 731-799. | MR 1354289 | Zbl 0851.53032

[7] M. Björklund, Central limit theorems for Gromov hyperbolic groups, J. Theoret. Probab. 23 (2010), 871-887. | MR 2679960 | Zbl 1217.60019

[8] S. Blachère & S. Brofferio, Internal diffusion limited aggregation on discrete groups having exponential growth, Probab. Theory Related Fields 137 (2007), 323-343. | MR 2278460 | Zbl 1106.60078

[9] S. Blachère, P. Haïssinsky & P. Mathieu, Asymptotic entropy and Green speed for random walks on countable groups, Ann. Probab. 36 (2008), 1134-1152. | MR 2408585 | Zbl 1146.60008

[10] M. Bonk & O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), 266-306. | MR 1771428 | Zbl 0972.53021

[11] M. Bourdon & H. Pajot, Quasi-conformal geometry and hyperbolic geometry, in Rigidity in dynamics and geometry (Cambridge, 2000), Springer, 2002, 1-17. | MR 1919393 | Zbl 1002.30012

[12] R. Bowen & C. Series, Markov maps associated with Fuchsian groups, Publ. Math. I.H.É.S. 50 (1979), 153-170. | Numdam | MR 556585 | Zbl 0439.30033

[13] C. Connell & R. Muchnik, Harmonicity of quasiconformal measures and Poisson boundaries of hyperbolic spaces, Geom. Funct. Anal. 17 (2007), 707-769. | MR 2346273 | Zbl 1166.60323

[14] M. Coornaert, Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov, Pacific J. Math. 159 (1993), 241-270. | MR 1214072 | Zbl 0797.20029

[15] B. Deroin, V. Kleptsyn & A. Navas, On the question of ergodicity for minimal group actions on the circle, Mosc. Math. J. 9 (2009), 263-303. | MR 2568439 | Zbl 1193.37034

[16] E. B. Dynkin, The boundary theory of Markov processes (discrete case), Uspehi Mat. Nauk 24 (1969), 3-42. | MR 245096 | Zbl 0222.60048

[17] E. Ghys & P. De La Harpe (éds.), Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Math. 83, Birkhäuser, 1990. | MR 1086648 | Zbl 0731.20025

[18] Y. Guivarc'H, Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire, in Conference on Random Walks (Kleebach, 1979), Astérisque 74, Soc. Math. France, 1980, 47-98. | MR 588157 | Zbl 0448.60007

[19] Y. Guivarc'H & Y. Le Jan, Sur l'enroulement du flot géodésique, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 645-648. | MR 1081425 | Zbl 0727.58033

[20] Y. Guivarc'H & Y. Le Jan, Asymptotic winding of the geodesic flow on modular surfaces and continued fractions, Ann. Sci. École Norm. Sup. 26 (1993), 23-50. | Numdam | MR 1209912 | Zbl 0784.60076

[21] J. Heinonen, Lectures on analysis on metric spaces, Universitext, Springer, 2001. | MR 1800917 | Zbl 0985.46008

[22] G. A. Hunt, Markoff chains and Martin boundaries, Illinois J. Math. 4 (1960), 313-340. | MR 123364 | Zbl 0094.32103

[23] V. A. Kaĭmanovich, Brownian motion and harmonic functions on covering manifolds. An entropic approach, Soviet Math. Dokl. 33 (1986), 812-816. | MR 852647 | Zbl 0615.60074

[24] V. A. Kaimanovich, Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Hyperbolic behaviour of dynamical systems, Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), 361-393. | Numdam | MR 1096098 | Zbl 0725.58026

[25] V. A. Kaimanovich, Discretization of bounded harmonic functions on Riemannian manifolds and entropy, in Potential theory (Nagoya, 1990), de Gruyter, 1992, 213-223. | MR 1167237 | Zbl 0768.58054

[26] V. A. Kaimanovich, Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces, J. reine angew. Math. 455 (1994), 57-103. | MR 1293874 | Zbl 0803.58032

[27] V. A. Kaimanovich, Hausdorff dimension of the harmonic measure on trees, Ergodic Theory Dynam. Systems 18 (1998), 631-660. | MR 1631732 | Zbl 0960.60047

[28] V. A. Kaimanovich, The Poisson formula for groups with hyperbolic properties, Ann. of Math. 152 (2000), 659-692. | MR 1815698 | Zbl 0984.60088

[29] I. Kapovich & N. Benakli, Boundaries of hyperbolic groups, in Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), Contemp. Math. 296, Amer. Math. Soc., 2002, 39-93. | MR 1921706 | Zbl 1044.20028

[30] A. Karlsson & F. Ledrappier, Propriété de Liouville et vitesse de fuite du mouvement brownien, C. R. Math. Acad. Sci. Paris 344 (2007), 685-690. | MR 2334676 | Zbl 1122.60071

[31] V. Le Prince, Dimensional properties of the harmonic measure for a random walk on a hyperbolic group, Trans. Amer. Math. Soc. 359 (2007), 2881-2898. | MR 2286061 | Zbl 1126.60036

[32] V. Le Prince, A relation between dimension of the harmonic measure, entropy and drift for a random walk on a hyperbolic space, Electron. Commun. Probab. 13 (2008), 45-53. | MR 2386061 | Zbl 1189.60094

[33] F. Ledrappier, Ergodic properties of Brownian motion on covers of compact negatively-curve manifolds, Bol. Soc. Brasil. Mat. 19 (1988), 115-140. | MR 1018929 | Zbl 0685.58036

[34] F. Ledrappier, Harmonic measures and Bowen-Margulis measures, Israel J. Math. 71 (1990), 275-287. | MR 1088820 | Zbl 0728.53029

[35] F. Ledrappier, Some asymptotic properties of random walks on free groups, in Topics in probability and Lie groups: boundary theory, CRM Proc. Lecture Notes 28, Amer. Math. Soc., 2001, 117-152. | MR 1832436 | Zbl 0994.60073

[36] R. Lyons, Equivalence of boundary measures on covering trees of finite graphs, Ergodic Theory Dynam. Systems 14 (1994), 575-597. | MR 1293410 | Zbl 0821.58008

[37] T. Lyons & D. Sullivan, Function theory, random paths and covering spaces, J. Differential Geom. 19 (1984), 299-323. | MR 755228 | Zbl 0554.58022

[38] J. Mairesse & F. Mathéus, Random walks on free products of cyclic groups, J. Lond. Math. Soc. 75 (2007), 47-66. | MR 2302729 | Zbl 1132.60054

[39] L. Naïm, Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel, Ann. Inst. Fourier, Grenoble 7 (1957), 183-281. | Numdam | MR 100174 | Zbl 0086.30603

[40] M. A. Pinsky, Stochastic Riemannian geometry, in Probabilistic analysis and related topics, Vol. 1, Academic Press, 1978, 199-236. | MR 501385 | Zbl 0452.60083

[41] J.-J. Prat, Étude asymptotique et convergence angulaire du mouvement brownien sur une variété à courbure négative, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), AA1539-A1542. | MR 388557 | Zbl 0309.60052

[42] J. Väisälä, Gromov hyperbolic spaces, Expo. Math. 23 (2005), 187-231. | MR 2164775 | Zbl 1087.53039

[43] A. M. Vershik, Dynamic theory of growth in groups: entropy, boundaries, examples, Russian Math. Surveys 55 (2000), 667-733. | MR 1786730 | Zbl 0991.37005

[44] W. Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics 138, Cambridge Univ. Press, 2000. | MR 1743100 | Zbl 0951.60002

[45] L. S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems 2 (1982), 109-124. | MR 684248 | Zbl 0523.58024