The Annals of Probability

Martin boundary of random walks with unbounded jumps in hyperbolic groups

Sébastien Gouëzel

Full-text: Open access


Given a probability measure on a finitely generated group, its Martin boundary is a natural way to compactify the group using the Green function of the corresponding random walk. For finitely supported measures in hyperbolic groups, it is known since the work of Ancona and Gouëzel–Lalley that the Martin boundary coincides with the geometric boundary. The goal of this paper is to weaken the finite support assumption. We first show that, in any nonamenable group, there exist probability measures with exponential tails giving rise to pathological Martin boundaries. Then, for probability measures with superexponential tails in hyperbolic groups, we show that the Martin boundary coincides with the geometric boundary by extending Ancona’s inequalities. We also deduce asymptotics of transition probabilities for symmetric measures with superexponential tails.

Article information

Ann. Probab., Volume 43, Number 5 (2015), 2374-2404.

Received: September 2013
First available in Project Euclid: 9 September 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 31C35: Martin boundary theory [See also 60J50] 60J50: Boundary theory 60B99: None of the above, but in this section

Random walk hyperbolic group Martin boundary Gromov boundary infinite range local limit theorem


Gouëzel, Sébastien. Martin boundary of random walks with unbounded jumps in hyperbolic groups. Ann. Probab. 43 (2015), no. 5, 2374--2404. doi:10.1214/14-AOP938.

Export citation


  • [1] Ancona, A. (1988). Positive harmonic functions and hyperbolicity. In Potential Theory—Surveys and Problems (Prague, 1987). Lecture Notes in Math. 1344 1–23. Springer, Berlin.
  • [2] Anderson, M. T. and Schoen, R. (1985). Positive harmonic functions on complete manifolds of negative curvature. Ann. of Math. (2) 121 429–461.
  • [3] Blachère, S., Haïssinsky, P. and Mathieu, P. (2011). Harmonic measures versus quasiconformal measures for hyperbolic groups. Ann. Sci. Éc. Norm. Supér. (4) 44 683–721.
  • [4] Bonk, M. and Schramm, O. (2000). Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. 10 266–306.
  • [5] Dynkin, E. B. (1969). The boundary theory of Markov processes (discrete case). Uspehi Mat. Nauk 24 3–42.
  • [6] Ghys, É. and de la Harpe, P., eds. (1990). Sur les Groupes Hyperboliques D’après Mikhael Gromov. Progress in Mathematics 83. Birkhäuser, Boston, MA.
  • [7] Gouëzel, S. (2014). Local limit theorem for symmetric random walks in Gromov-hyperbolic groups. J. Amer. Math. Soc. 27 893–928.
  • [8] Gouëzel, S. and Lalley, S. P. (2013). Random walks on co-compact Fuchsian groups. Ann. Sci. Éc. Norm. Supér. (4) 46 129–173.
  • [9] Izumi, M., Neshveyev, S. and Okayasu, R. (2008). The ratio set of the harmonic measure of a random walk on a hyperbolic group. Israel J. Math. 163 285–316.
  • [10] Ledrappier, F. (2001). Some asymptotic properties of random walks on free groups. In Topics in Probability and Lie Groups: Boundary Theory. CRM Proc. Lecture Notes 28 117–152. Amer. Math. Soc., Providence, RI.
  • [11] Sawyer, S. A. (1997). Martin boundaries and random walks. In Harmonic Functions on Trees and Buildings (New York, 1995). Contemp. Math. 206 17–44. Amer. Math. Soc., Providence, RI.
  • [12] Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138. Cambridge Univ. Press, Cambridge.