Harmonic measures versus quasiconformal measures for hyperbolic groups

Sébastien Blachère; Peter Haïssinsky; Pierre Mathieu

Annales scientifiques de l'École Normale Supérieure (2011)

  • Volume: 44, Issue: 4, page 683-721
  • ISSN: 0012-9593

Abstract

top
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.

How to cite

top

Blachère, Sébastien, Haïssinsky, Peter, and Mathieu, Pierre. "Harmonic measures versus quasiconformal measures for hyperbolic groups." Annales scientifiques de l'École Normale Supérieure 44.4 (2011): 683-721. <http://eudml.org/doc/272118>.

@article{Blachère2011,
abstract = {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.},
author = {Blachère, Sébastien, Haïssinsky, Peter, Mathieu, Pierre},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {hyperbolic groups; random walks on groups; harmonic measures; quasiconformal measures; dimension of a measure; Martin boundary; brownian motion; Green metric},
language = {eng},
number = {4},
pages = {683-721},
publisher = {Société mathématique de France},
title = {Harmonic measures versus quasiconformal measures for hyperbolic groups},
url = {http://eudml.org/doc/272118},
volume = {44},
year = {2011},
}

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
PB - Société mathématique de France
VL - 44
IS - 4
SP - 683
EP - 721
AB - 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.
LA - eng
KW - hyperbolic groups; random walks on groups; harmonic measures; quasiconformal measures; dimension of a measure; Martin boundary; brownian motion; Green metric
UR - http://eudml.org/doc/272118
ER -

References

top
  1. [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. Zbl0677.31006MR973878
  2. [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. Zbl0719.60074MR1100282
  3. [3] M. T. Anderson & R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math.121 (1985), 429–461. Zbl0587.53045MR794369
  4. [4] W. Ballmann, On the Dirichlet problem at infinity for manifolds of nonpositive curvature, Forum Math.1 (1989), 201–213. Zbl0661.53026MR990144
  5. [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. Zbl0885.53037MR1427756
  6. [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. Zbl0851.53032MR1354289
  7. [7] M. Björklund, Central limit theorems for Gromov hyperbolic groups, J. Theoret. Probab.23 (2010), 871–887. Zbl1217.60019MR2679960
  8. [8] S. Blachère & S. Brofferio, Internal diffusion limited aggregation on discrete groups having exponential growth, Probab. Theory Related Fields137 (2007), 323–343. Zbl1106.60078MR2278460
  9. [9] S. Blachère, P. Haïssinsky & P. Mathieu, Asymptotic entropy and Green speed for random walks on countable groups, Ann. Probab.36 (2008), 1134–1152. Zbl1146.60008MR2408585
  10. [10] M. Bonk & O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal.10 (2000), 266–306. Zbl0972.53021MR1771428
  11. [11] M. Bourdon & H. Pajot, Quasi-conformal geometry and hyperbolic geometry, in Rigidity in dynamics and geometry (Cambridge, 2000), Springer, 2002, 1–17. Zbl1002.30012MR1919393
  12. [12] R. Bowen & C. Series, Markov maps associated with Fuchsian groups, Publ. Math. I.H.É.S. 50 (1979), 153–170. Zbl0439.30033MR556585
  13. [13] C. Connell & R. Muchnik, Harmonicity of quasiconformal measures and Poisson boundaries of hyperbolic spaces, Geom. Funct. Anal.17 (2007), 707–769. Zbl1166.60323MR2346273
  14. [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. Zbl0797.20029MR1214072
  15. [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. Zbl1193.37034MR2568439
  16. [16] E. B. Dynkin, The boundary theory of Markov processes (discrete case), Uspehi Mat. Nauk24 (1969), 3–42. Zbl0222.60048MR245096
  17. [17] E. Ghys & P. de la Harpe (éds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Math. 83, Birkhäuser, 1990. Zbl0731.20025MR1086648
  18. [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. Zbl0448.60007MR588157
  19. [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. Zbl0727.58033MR1081425
  20. [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. Zbl0784.60076MR1209912
  21. [21] J. Heinonen, Lectures on analysis on metric spaces, Universitext, Springer, 2001. Zbl0985.46008MR1800917
  22. [22] G. A. Hunt, Markoff chains and Martin boundaries, Illinois J. Math.4 (1960), 313–340. Zbl0094.32103MR123364
  23. [23] V. A. Kaĭmanovich, Brownian motion and harmonic functions on covering manifolds. An entropic approach, Soviet Math. Dokl. 33 (1986), 812–816. Zbl0615.60074MR852647
  24. [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. Zbl0725.58026MR1096098
  25. [25] V. A. Kaimanovich, Discretization of bounded harmonic functions on Riemannian manifolds and entropy, in Potential theory (Nagoya, 1990), de Gruyter, 1992, 213–223. Zbl0768.58054MR1167237
  26. [26] V. A. Kaimanovich, Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces, J. reine angew. Math. 455 (1994), 57–103. Zbl0803.58032MR1293874
  27. [27] V. A. Kaimanovich, Hausdorff dimension of the harmonic measure on trees, Ergodic Theory Dynam. Systems18 (1998), 631–660. Zbl0960.60047MR1631732
  28. [28] V. A. Kaimanovich, The Poisson formula for groups with hyperbolic properties, Ann. of Math.152 (2000), 659–692. Zbl0984.60088MR1815698
  29. [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. Zbl1044.20028MR1921706
  30. [30] A. Karlsson & F. Ledrappier, Propriété de Liouville et vitesse de fuite du mouvement brownien, C. R. Math. Acad. Sci. Paris344 (2007), 685–690. Zbl1122.60071MR2334676
  31. [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. Zbl1126.60036MR2286061
  32. [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. Zbl1189.60094MR2386061
  33. [33] F. Ledrappier, Ergodic properties of Brownian motion on covers of compact negatively-curve manifolds, Bol. Soc. Brasil. Mat.19 (1988), 115–140. Zbl0685.58036MR1018929
  34. [34] F. Ledrappier, Harmonic measures and Bowen-Margulis measures, Israel J. Math.71 (1990), 275–287. Zbl0728.53029MR1088820
  35. [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. Zbl0994.60073MR1832436
  36. [36] R. Lyons, Equivalence of boundary measures on covering trees of finite graphs, Ergodic Theory Dynam. Systems14 (1994), 575–597. Zbl0821.58008MR1293410
  37. [37] T. Lyons & D. Sullivan, Function theory, random paths and covering spaces, J. Differential Geom.19 (1984), 299–323. Zbl0554.58022MR755228
  38. [38] J. Mairesse & F. Mathéus, Random walks on free products of cyclic groups, J. Lond. Math. Soc.75 (2007), 47–66. Zbl1132.60054MR2302729
  39. [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. Zbl0086.30603MR100174
  40. [40] M. A. Pinsky, Stochastic Riemannian geometry, in Probabilistic analysis and related topics, Vol. 1, Academic Press, 1978, 199–236. Zbl0452.60083MR501385
  41. [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. Zbl0309.60052MR388557
  42. [42] J. Väisälä, Gromov hyperbolic spaces, Expo. Math.23 (2005), 187–231. Zbl1087.53039MR2164775
  43. [43] A. M. Vershik, Dynamic theory of growth in groups: entropy, boundaries, examples, Russian Math. Surveys55 (2000), 667–733. Zbl0991.37005MR1786730
  44. [44] W. Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics 138, Cambridge Univ. Press, 2000. Zbl0951.60002MR1743100
  45. [45] L. S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems2 (1982), 109–124. Zbl0523.58024MR684248

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.