The Liouville property for harmonic functions on groups and hypergroups

Authors

  • H. Heyer Mathematisches Institut, Universitat Tubingen, Auf der Morgenstelle 10, D-72076 Tubingen, Germany

DOI:

Keywords:

Harmonic functions, information measures, hypergroups

Abstract

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.

References

Massoud Amini, Harmonic functions on [IN] and central hypergroups, Monatsh. Math. 169 (2013), no. 3-4, 267-284. MathSciNet CrossRef

Massoud Amini and Cho-Ho Chu, Harmonic functions on hypergroups, J. Funct. Anal. 261 (2011), no. 7, 1835-1864. MathSciNet CrossRef

A. Avez, Entropie des groupes de type fini, C. R. Acad. Sci. Paris Ser. A-B 275 (1972), A1363-A1366. MathSciNet

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

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

Robert Azencott, Espaces de Poisson des groupes localement compacts, Lecture Notes in Mathematics, Vol. 148, Springer-Verlag, Berlin-New York, 1970. MathSciNet

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

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

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

Gustave Choquet and Jacques Deny, Sur lequation de convolution $mu =mu ast sigma $, C. R. Acad. Sci. Paris 250 (1960), 799-801. MathSciNet

Cho-Ho Chu, Harmonic function spaces on groups, J. London Math. Soc. (2) 70 (2004), no. 1, 182-198. MathSciNet CrossRef

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

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

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

Cho-Ho Chu and Chi-Wai Leung, Harmonic functions on homogeneous spaces, Monatsh. Math. 128 (1999), no. 3, 227-235. MathSciNet CrossRef

Y. Derriennic, Lois ``zero ou deux pour les processus de Markov. Applications aux marches aleatoires, Ann. Inst. H. Poincare Sect. B (N.S.) 12 (1976), no. 2, 111-129. MathSciNet

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

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

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

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

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

Anna Erschler, Liouville property for groups and manifolds, Invent. Math. 155 (2004), no. 1, 55-80. MathSciNet CrossRef

Harry Furstenberg, Noncommuting random products, Trans. Amer. Math. Soc. 108 (1963), 377-428. MathSciNet

Harry Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963), 335-386. MathSciNet

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

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

Shmuel Glasner, Proximal flows, Lecture Notes in Mathematics, Vol. 517, Springer-Verlag, Berlin-New York, 1976. MathSciNet

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

Yves Guivarch, Croissance polynomiale et periodes des fonctions harmoniques, Bull. Soc. Math. France 101 (1973), 333-379. MathSciNet

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

H. Heyer, Potentialtheorie auf Lie-Gruppen, (1967),

H. Heyer, Information functionals with applications to random walk and statistics, J. Stat. Theory Pract. 9 (2015), no. 4, 896-933. MathSciNet CrossRef

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

B. E. Johnson, Harmonic functions on nilpotent groups, Integral Equations Operator Theory 40 (2001), no. 4, 454-464. MathSciNet CrossRef

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

V. A. Kaimanovich, Amenability and the Liouville property, Israel J. Math. 149 (2005), 45-85. MathSciNet CrossRef

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

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

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

Albert Raugi, Fonctions harmoniques positives sur certains groupes de Lie resolubles connexes, Bull. Soc. Math. France 124 (1996), no. 4, 649-684. MathSciNet

D. Revuz, Markov chains, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MathSciNet

Joseph Rosenblatt, Ergodic and mixing random walks on locally compact groups, Math. Ann. 257 (1981), no. 1, 31-42. MathSciNet CrossRef

Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Dusseldorf-Johannesburg, 1974. MathSciNet

Downloads

Published

2017-03-25

Issue

Section

Articles

How to Cite

Heyer, H. “The Liouville Property for Harmonic Functions on Groups and Hypergroups”. Methods of Functional Analysis and Topology, vol. 23, no. 1, Mar. 2017, pp. 3-25, https://zen.imath.kiev.ua/index.php/mfat/article/view/650.