Positive harmonic functions on the Heisenberg group II
[Fonctions harmoniques positives sur le groupe de Heisenberg II]
Benoist, Yves1
1 IMO, CNRS, Université Paris-Saclay Bâtiment 307, 91405 Orsay, France
Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 973-1003

We describe the extremal positive harmonic functions for finitely supported measures on the discrete Heisenberg group: they are proportional either to characters or to translates of induced from characters.

Nous décrivons les fonctions harmoniques positives extrémales pour les mesures à support fini sur le groupe de Heisenberg discret : elles sont proportionnelles à des caractères ou à des translatées d’induites de caractères.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.163
Classification : 31C35, 60B15, 60G50, 60J50
Keywords: Harmonic function, Martin boundary, random walk, nilpotent group
Mots-clés : Fonction harmonique, marche aléatoire, frontière de Martin, groupe nilpotent

Benoist, Yves 1

1 IMO, CNRS, Université Paris-Saclay Bâtiment 307, 91405 Orsay, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{JEP_2021__8__973_0,
     author = {Benoist, Yves},
     title = {Positive harmonic functions on {the~Heisenberg} group {II}},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {973--1003},
     year = {2021},
     publisher = {Ecole polytechnique},
     volume = {8},
     doi = {10.5802/jep.163},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/jep.163/}
}
TY  - JOUR
AU  - Benoist, Yves
TI  - Positive harmonic functions on the Heisenberg group II
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2021
SP  - 973
EP  - 1003
VL  - 8
PB  - Ecole polytechnique
UR  - https://www.numdam.org/articles/10.5802/jep.163/
DO  - 10.5802/jep.163
LA  - en
ID  - JEP_2021__8__973_0
ER  - 
%0 Journal Article
%A Benoist, Yves
%T Positive harmonic functions on the Heisenberg group II
%J Journal de l’École polytechnique — Mathématiques
%D 2021
%P 973-1003
%V 8
%I Ecole polytechnique
%U https://www.numdam.org/articles/10.5802/jep.163/
%R 10.5802/jep.163
%G en
%F JEP_2021__8__973_0
Benoist, Yves. Positive harmonic functions on the Heisenberg group II. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 973-1003. doi: 10.5802/jep.163

[1] Ancona, A. Théorie du potentiel sur les graphes et les variétés, École d’été de Probabilités de Saint-Flour XVIII—1988 (Lect. Notes in Math.), Volume 1427, Springer, Berlin, 1990, pp. 1-112 | Zbl | DOI

[2] Benoist, Yves Positive harmonic functions on the Heisenberg group I, 2019 | arXiv

[3] Breuillard, Emmanuel Local limit theorems and equidistribution of random walks on the Heisenberg group, Geom. Funct. Anal., Volume 15 (2005) no. 1, pp. 35-82 | Zbl | MR | DOI

[4] Breuillard, Emmanuel Equidistribution of dense subgroups on nilpotent Lie groups, Ergodic Theory Dynam. Systems, Volume 30 (2010) no. 1, pp. 131-150 | Zbl | MR | DOI

[5] Choquet, Gustave; Deny, Jacques Sur l’équation de convolution μ=μ*σ, C. R. Acad. Sci. Paris, Volume 250 (1960), pp. 799-801 | Zbl

[6] Diaconis, Persi; Hough, Bob Random walk on unipotent matrix groups, 2015 | arXiv

[7] Dynkin, E. B.; Maljutov, M. B. Random walk on groups with a finite number of generators, Dokl. Akad. Nauk SSSR, Volume 137 (1961), pp. 1042-1045 | MR

[8] Göll, Martin; Schmidt, Klaus; Verbitskiy, Evgeny A Wiener lemma for the discrete Heisenberg group, Monatsh. Math., Volume 180 (2016) no. 3, pp. 485-525 | MR | DOI

[9] Gouëzel, Sébastien Martin boundary of random walks with unbounded jumps in hyperbolic groups, Ann. Probab., Volume 43 (2015) no. 5, pp. 2374-2404 | Zbl | MR | DOI

[10] Guivarc’h, Yves Groupes nilpotents et probabilité, C. R. Acad. Sci. Paris Sér. A-B, Volume 273 (1971), p. A997-A998 | Zbl

[11] Lind, D.; Shmidt, K. A survey of algebraic actions of the discrete Heisenberg group, Uspekhi Mat. Nauk, Volume 70 (2015) no. 4(424), pp. 77-142 | MR | DOI

[12] Margulis, G. A. Positive harmonic functions on nilpotent groups, Soviet Math. Dokl., Volume 7 (1966), pp. 241-244 | Zbl | MR

[13] Sawyer, Stanley A. Martin boundaries and random walks, Harmonic functions on trees and buildings (New York, 1995) (Contemp. Math.), Volume 206, American Mathematical Society, Providence, RI, 1997, pp. 17-44 | Zbl | MR | DOI

[14] Vershik, A. M.; Malyutin, A. V. Asymptotics of the number of geodesics in the discrete Heisenberg group, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Volume 468 (2018) no. XXIX, pp. 39-52

[15] Woess, Wolfgang Random walks on infinite graphs and groups, Cambridge Tracts in Math., 138, Cambridge University Press, Cambridge, 2000 | Zbl | MR | DOI

Cité par Sources :