Random walks of infinite moment on free semigroups

  • Behrang Forghani
  • Giulio TiozzoEmail author


We consider random walks on finitely or countably generated free semigroups, and identify their Poisson boundaries for classes of measures which fail to meet the classical entropy criteria, namely measures with infinite entropy or infinite logarithmic moment.


Random walks Poisson boundary Free semigroups Free groups 

Mathematics Subject Classification

60G50 60J50 05C81 



We would like to thank Lewis Bowen, Vadim Kaimanovich, and Joseph Maher for fruitful discussions. G. T. is partially supported by NSERC and the Alfred P. Sloan Foundation.


  1. 1.
    Ancona, A.: Positive harmonic functions and hyperbolicity. In: Potential Theory—Surveys and Problems (Prague, 1987), Volume 1344 of Lecture Notes in Mathematics, pp. 1–23. Springer, Berlin (1988)Google Scholar
  2. 2.
    Blackwell, D.: On transient Markov processes with a countable number of states and stationary transition probabilities. Ann. Math. Stat. 26, 654–658 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brooks, R.: Some remarks on bounded cohomology. In: Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (SUNY, Stony Brook, NY, 1978) (Annals of Mathematics Studies), vol. 97, pp. 53–63. Princeton University Press, Princeton (1981)Google Scholar
  4. 4.
    Choquet, G., Deny, J.: Sur l’équation de convolution \(\mu =\mu \ast \sigma \). C. R. Acad. Sci. Paris 250, 799–801 (1960)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Calegari, D., Maher, J.: Statistics and compression of scl. Ergod. Theory Dyn. Syst. 35(1), 64–110 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Derriennic, Y.: Quelques applications du théorème ergodique sous-additif. In: Conference on Random Walks (Kleebach, 1979) (French), Volume 74 of Astérisque, vol. 4, pp. 183–201. Société Mathématique de France, Paris (1980)Google Scholar
  7. 7.
    Dynkin, E.B., Maljutov, M.B.: Random walk on groups with a finite number of generators. Dokl. Akad. Nauk SSSR 137, 1042–1045 (1961)MathSciNetGoogle Scholar
  8. 8.
    Erschler, A.: Poisson–Furstenberg boundaries, large-scale geometry and growth of groups. In: Proceedings of the International Congress of Mathematicians, vol. II, pp. 681–704. Hindustan Book Agency, New Delhi (2010)Google Scholar
  9. 9.
    Feller, W.: Boundaries induced by non-negative matrices. Trans. Am. Math. Soc. 83, 19–54 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Forghani, B., Kaimanovich, V.A.: Boundary preserving transformations of random walks (2017). (in Preparation) Google Scholar
  11. 11.
    Forghani, B.: Transformed Random Walks. Ph.D. thesis, University of Ottawa, Canada (2015)Google Scholar
  12. 12.
    Forghani, B.: Asymptotic entropy of transformed random walks. Ergod. Theory Dyn. Syst. 37, 1480–1491 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Furstenberg, H.: Random walks and discrete subgroups of Lie groups. In: Advances in Probability and Related Topics, vol. 1, pp. 1–63. Dekker, New York (1971)Google Scholar
  14. 14.
    Furman, A.: Random walks on groups and random transformations. In: Handbook of Dynamical Systems, vol. 1A, pp. 931–1014. North-Holland, Amsterdam (2002)Google Scholar
  15. 15.
    Gautero, F., Mathéus, F.: Poisson boundary of groups acting on \({\mathbb{R}}\)-trees. Isr. J. Math. 191(2), 585–646 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gouëzel, S.: Martin boundary of random walks with unbounded jumps in hyperbolic groups. Ann. Probab. 43(5), 2374–2404 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hanson, D.L.: Limiting sigma-algebras—some counter examples. In: Asymptotic Methods in Probability and Statistics (Ottawa, ON, 1997), pp. 383–385. North-Holland, Amsterdam (1998)Google Scholar
  18. 18.
    Horbez, C.: The Poisson boundary of \(Out(F_N)\). Duke Math. J. 165(2), 341–369 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kaimanovich, V.: An entropy criterion for maximality of the boundary of random walks on discrete groups. Sov. Math. Dokl. 31, 193–197 (1985)Google Scholar
  20. 20.
    Kaimanovich, V.A.: The poisson boundary of hyperbolic groups. Comptes rendus de l’Académie des sciences. Série 1, Mathématique 318(1), 59–64 (1994)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kaimanovich, V.A.: The Poisson formula for groups with hyperbolic properties. Ann. Math. (2) 152(3), 659–692 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kaimanovich, V., Masur, H.: The poisson boundary of the mapping class group. Invent. math. 125, 221–264 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Karlsson, A., Margulis, G.A.: A multiplicative ergodic theorem and nonpositively curved spaces. Commun. Math. Phys. 208(1), 107–123 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kaimanovich, V.A., Sobieczky, F.: Random walks on random horospheric products. In: Dynamical Systems and Group Actions, Volume 567 of Contemporary Mathematics, pp. 163–183. American Mathematical Society, Providence (2012)Google Scholar
  25. 25.
    Kaimanovich, V.A., Vershik, A.M.: Random walks on discrete groups: boundary and entropy. Ann. Probab. 11(3), 457–490 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Karlsson, A., Woess, W.: The Poisson boundary of lamplighter random walks on trees. Geom. Dedicata 124, 95–107 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ledrappier, F.: Some asymptotic properties of random walks on free groups. In: Topics in Probability and Lie Groups: Boundary Theory, Volume 28 of CRM Proceedings and Lecture Notes, pp. 117–152. American Mathematical Society, Providence (2001)Google Scholar
  28. 28.
    Maher, J.: Random walks on the mapping class group. Duke Math. J. 156(3), 429–468 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Maher, J., Tiozzo, G.: Random walks on weakly hyperbolic groups. Journal für die reine und angewandte Mathematik (Crelles Journal) 742, 187–239 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Rohlin, V.A.: On the fundamental ideas of measure theory. Am. Math. Soc. Transl. 1952(71), 55 (1952)MathSciNetGoogle Scholar
  31. 31.
    Woess, W.: Denumerable Markov Chains: Generating Functions, Boundary Theory, Random Walks on Trees. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich (2009)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsBowdoin CollegeBrunswickUSA
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada

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