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The Liouville property for harmonic functions on groups and hypergroups


A survey is given on the Liouville property of harmonic functions on groups and hypergroups. The discussion of a characterization of that property in terms of the underlying algebraic structures yields interesting open problems.

Key words: Harmonic functions, information measures, hypergroups.

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TitleThe Liouville property for harmonic functions on groups and hypergroups
SourceMethods Funct. Anal. Topology, Vol. 23 (2017), no. 1, 3-25
MathSciNet   MR3632385
zbMATH 06810664
Milestones  Received 04/10/2016; Revised 19/12/2016
CopyrightThe Author(s) 2017 (CC BY-SA)

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Herbert Heyer
Mathematisches Institut, Universitat Tubingen, Auf der Morgenstelle 10, D-72076 Tubingen, Germany

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Herbert Heyer, The Liouville property for harmonic functions on groups and hypergroups, Methods Funct. Anal. Topology 23 (2017), no. 1, 3-25.


@article {MFAT945,
    AUTHOR = {Heyer, Herbert},
     TITLE = {The Liouville property for harmonic functions on groups and hypergroups},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {23},
      YEAR = {2017},
    NUMBER = {1},
     PAGES = {3-25},
      ISSN = {1029-3531},
  MRNUMBER = {MR3632385},
 ZBLNUMBER = {06810664},
       URL = {},


  1. Massoud Amini, Harmonic functions on [IN] and central hypergroups, Monatsh. Math. 169 (2013), no. 3-4, 267-284.  MathSciNet CrossRef
  2. Massoud Amini and Cho-Ho Chu, Harmonic functions on hypergroups, J. Funct. Anal. 261 (2011), no. 7, 1835-1864.  MathSciNet CrossRef
  3. A. Avez, Entropie des groupes de type fini, C. R. Acad. Sci. Paris S\er. A-B 275 (1972), A1363-A1366.  MathSciNet
  4. A. Avez, Croissance des groupes de type fini et fonctions harmoniques, Probability measures on groups, Lecture Notes in Math., vol. 532, Springer, Berlin, 1976, pp. 35-49..  MathSciNet CrossRef
  5. A. Avez, Harmonic functions on groups, Differential geometry and relativity. Mathematical Phys. and Appl. Math., Vol. 3, Reidel, Dordrecht, 1976, pp. 27-32.  MathSciNet CrossRef
  6. Robert Azencott, Espaces de Poisson des groupes localement compacts, Lecture Notes in Mathematics, Vol. 148, Springer-Verlag, Berlin-New York, 1970.  MathSciNet
  7. M. Babillot, An introduction to Poisson boundaries of Lie groups, Probability measures on groups: recent directions and trends, Tata Inst. Fund. Res., Mumbai, 2006, pp. 1-90.  MathSciNet
  8. Yu. M. Berezansky and A. A. Kalyuzhnyi, Harmonic analysis in hypercomplex systems, Mathematics and its Applications, vol. 434, Kluwer Academic Publishers, Dordrecht, 1998.  MathSciNet CrossRef
  9. Walter R. Bloom and Herbert Heyer, Harmonic analysis of probability measures on hypergroups, de Gruyter Studies in Mathematics, vol. 20, Walter de Gruyter & Co., Berlin, 1995.  MathSciNet CrossRef
  10. Gustave Choquet and Jacques Deny, Sur lequation de convolution $\mu =\mu \ast \sigma $, C. R. Acad. Sci. Paris 250 (1960), 799-801.  MathSciNet
  11. Cho-Ho Chu, Harmonic function spaces on groups, J. London Math. Soc. (2) 70 (2004), no. 1, 182-198.  MathSciNet CrossRef
  12. Cho-Ho Chu and Titus Hilberdink, The convolution equation of Choquet and Deny on nilpotent groups, Integral Equations Operator Theory 26 (1996), no. 1, 1-13.  MathSciNet CrossRef
  13. Cho-Ho Chu and Anthony To-Ming Lau, Harmonic functions on groups and Fourier algebras, Lecture Notes in Mathematics, vol. 1782, Springer-Verlag, Berlin, 2002. CrossRef
  14. Cho-Ho Chu and Anthony To-Ming Lau, Harmonic functions on topological groups and symmetric spaces, Math. Z. 268 (2011), no. 3-4, 649-673.  MathSciNet CrossRef
  15. Cho-Ho Chu and Chi-Wai Leung, Harmonic functions on homogeneous spaces, Monatsh. Math. 128 (1999), no. 3, 227-235.  MathSciNet CrossRef
  16. Y. Derriennic, Lois ``zero ou deux pour les processus de Markov. Applications aux marches aleatoires, Ann. Inst. H. Poincar\e Sect. B (N.S.) 12 (1976), no. 2, 111-129.  MathSciNet
  17. Y. Derriennic, Entropie, theor\`emes limite et marches aleatoires, Probability measures on groups, VIII (Oberwolfach, 1985), Lecture Notes in Math., vol. 1210, Springer, Berlin, 1986, pp. 241-284.  MathSciNet CrossRef
  18. Y. Derriennic, Entropy and boundary for random walks on locally compact groups--the example of the affine group, Transactions of the Tenth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, Vol.\ A (Prague, 1986), Reidel, Dordrecht, 1988, pp. 269-275.  MathSciNet
  19. Y. Derriennic and M. Lin, Sur la tribu asymptotique des marches aleatoires sur les groupes, Seminaires de probabilites Rennes 1983, Publ. S\'em. Math., Univ. Rennes I, Rennes, 1983, pp. 8.  MathSciNet
  20. E. B. Dynkin and M. B. Maljutov, Random walk on groups with a finite number of generators, Dokl. Akad. Nauk SSSR 137 (1961), 1042-1045.  MathSciNet
  21. Laure \Elie, Fonctions harmoniques positives sur le groupe affine, Probability measures on groups, Lecture Notes in Math., vol. 706, Springer, Berlin, 1979, pp. 96-110.  MathSciNet CrossRef
  22. Anna Erschler, Liouville property for groups and manifolds, Invent. Math. 155 (2004), no. 1, 55-80.  MathSciNet CrossRef
  23. Harry Furstenberg, Noncommuting random products, Trans. Amer. Math. Soc. 108 (1963), 377-428.  MathSciNet
  24. Harry Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963), 335-386.  MathSciNet
  25. Harry Furstenberg, Random walks and discrete subgroups of Lie groups, Advances in Probability and Related Topics, Vol. 1, Dekker, New York, 1971, pp. 1-63.  MathSciNet
  26. Harry Furstenberg, Boundary theory and stochastic processes on homogeneous spaces, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972), Amer. Math. Soc., Providence, R.I., 1973, pp. 193-229.  MathSciNet
  27. Shmuel Glasner, Proximal flows, Lecture Notes in Mathematics, Vol. 517, Springer-Verlag, Berlin-New York, 1976.  MathSciNet
  28. Y. Guivarch, Sur la loi des grands nombres et le rayon spectral dune marche aleatoire, Conference on Random Walks (Kleebach, 1979) (French), Ast\'erisque, vol. 74, Soc. Math. France, Paris, 1980, pp. 47-98, 3.  MathSciNet
  29. Yves Guivarch, Croissance polynomiale et periodes des fonctions harmoniques, Bull. Soc. Math. France 101 (1973), 333-379.  MathSciNet
  30. Wilfried Hauenschild, Eberhard Kaniuth, and Ajay Kumar, Harmonic analysis on central hypergroups and induced representations, Pacific J. Math. 110 (1984), no. 1, 83-112.  MathSciNet
  31. H. Heyer, Potentialtheorie auf Lie-Gruppen, (1967),
  32. H. Heyer, Information functionals with applications to random walk and statistics, J. Stat. Theory Pract. 9 (2015), no. 4, 896-933.  MathSciNet CrossRef
  33. Wojciech Jaworski, On the asymptotic and invariant $\sigma$-algebras of random walks on locally compact groups, Probab. Theory Related Fields 101 (1995), no. 2, 147-171.  MathSciNet CrossRef
  34. B. E. Johnson, Harmonic functions on nilpotent groups, Integral Equations Operator Theory 40 (2001), no. 4, 454-464.  MathSciNet CrossRef
  35. V. A. Kaimanovich, Differential entropy of the boundary of a random walk on a group, Uspekhi Mat. Nauk 38 (1983), no. 5(233), 187-188.  MathSciNet CrossRef
  36. V. A. Kaimanovich, Amenability and the Liouville property, Israel J. Math. 149 (2005), 45-85.  MathSciNet CrossRef
  37. V. A. Kaimanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Probab. 11 (1983), no. 3, 457-490.  MathSciNet CrossRef
  38. Yukiyosi Kawada and Kiyosi Ito, On the probability distribution on a compact group. I, Proc. Phys.-Math. Soc. Japan (3) 22 (1940), 977-998.  MathSciNet
  39. M. S. Pinsker, Information and information stability of random variables and processes, Translated and edited by Amiel Feinstein, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964.  MathSciNet
  40. Albert Raugi, Fonctions harmoniques positives sur certains groupes de Lie resolubles connexes, Bull. Soc. Math. France 124 (1996), no. 4, 649-684.  MathSciNet
  41. D. Revuz, Markov chains, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975.  MathSciNet
  42. Joseph Rosenblatt, Ergodic and mixing random walks on locally compact groups, Math. Ann. 257 (1981), no. 1, 31-42.  MathSciNet CrossRef
  43. Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Dusseldorf-Johannesburg, 1974.  MathSciNet

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