no. 1
The Martin boundary of a free product of abelian groups
[Le bord de Martin d’un produit libre de groupes abéliens]
Dussaule, Matthieu1
1 Laboratoire Jean Leray 2 rue de la Houssinière BP92208 44322 Nantes Cedex 3 (France)
Annales de l'Institut Fourier, Tome 70 (2020) no. 1, pp. 313-373.

Étant donné une mesure de probabilité sur un groupe de type fini Γ, on définit son bord de Martin qui donne une manière de compactifier Γ à l’aide de la fonction de Green associée. On sait depuis les travaux de W. Woess que le bord de Martin d’une marche aléatoire adaptée sur un produit libre de groupes abéliens coïncide avec le bord géométrique. Le but principal de cet article est d’étendre ce résultat aux marches à support fini qui ne sont pas adaptées. Il n’y a alors pas de formule explicite pour la fonction de Green, mais on montre encore que le bord de Martin coïncide avec le bord géométrique. On prouve au passage que le bord de Martin est minimal.

Given a probability measure μ on a finitely generated group Γ, its Martin boundary is a way to compactify Γ using the Green function of the corresponding random walk. It is known from the work of W. Woess that when a finitely supported random walk on a free product of abelian groups is adapted to the free product structure, the Martin boundary coincides with the geometric boundary. The main goal of this paper is to deal with non-adapted finitely supported random walks, for which there is no explicit formula for the Green function. Nevertheless, we show that the Martin boundary still coincides with the geometric boundary. We also prove that the Martin boundary is minimal.

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Révisé le :
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DOI : 10.5802/aif.3314
Classification : 05C81, 60B15, 31C35, 20F67
Keywords: random walk, free product, Martin boundary, harmonic function
Mot clés : marche aléatoire, produit libre, bord de Martin, fonction harmonique

Dussaule, Matthieu 1

1 Laboratoire Jean Leray 2 rue de la Houssinière BP92208 44322 Nantes Cedex 3 (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Dussaule, Matthieu. The Martin boundary of a free product of abelian groups. Annales de l'Institut Fourier, Tome 70 (2020) no. 1, pp. 313-373. doi : 10.5802/aif.3314. https://aif.centre-mersenne.org/articles/10.5802/aif.3314/

[1] Alexopoulos, Georgios Random walks on discrete groups of polynomial volume growth, Ann. Probab., Volume 30 (2002), pp. 723-801 | MR | Zbl

[2] Ancona, Alano Positive harmonic functions and hyperbolicity, Potential theory-surveys and problems (Lecture Notes in Mathematics), Volume 1344, Springer, 1988, pp. 1-23 | DOI | MR | Zbl

[3] Ancona, Alano Théorie du potentiel sur les graphes et les variétés, École d’été de Probabilités de Saint-Flour XVIII (Lecture Notes in Mathematics), Volume 1427, Springer, 1990, pp. 1-112 | Zbl

[4] Babillot, Martine Théorie du renouvellement pour des chaînes semi-markoviennes transientes, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 24 (1988) no. 4, pp. 507-569 | Numdam | MR | Zbl

[5] Billingsley, Patrick Convergence of Probability Measures, John Wiley & Sons, 1999 | DOI | Zbl

[6] Birkhoff, Garrett Extension of Jentzsch’s theorem, Trans. Am. Math. Soc., Volume 85 (1957), pp. 219-227 | MR | Zbl

[7] Blachère, Sébastien; Brofferio, Sara Internal Diffusion Limited Aggregation on discrete groups having exponential growth, Probab. Theory Relat. Fields, Volume 137 (2007), pp. 323-343 | DOI | MR | Zbl

[8] Blachère, Sébastien; Haïssinsky, Peter; Mathieu, Pierre Harmonic measures versus quasiconformal measures for hyperbolic groups, Ann. Sci. Éc. Norm. Supér., Volume 44 (2011), pp. 683-721 | DOI | Numdam | MR | Zbl

[9] Bridson, Martin; Häfliger, André Metric Spaces of Non-Positive Curvature, Springer, 1999 | DOI | Zbl

[10] Cartwright, Donald; Sawyer, Stanley The Martin boundary for general isotropic random walks in a tree, J. Theor. Probab., Volume 4 (1991), pp. 111-136 | DOI | MR | Zbl

[11] Dahmani, François Classifying spaces and boundaries for relatively hyperbolic groups, Proc. Lond. Math. Soc., Volume 86 (2003), pp. 666-684 | DOI | MR | Zbl

[12] Derriennic, Yves Marche aléatoire sur le groupe libre et frontière de Martin, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 32 (1975), pp. 261-276 | DOI | Zbl

[13] Gekhtman, Ilya; Gerasimov, Victor; Potyagailo, Loenid; Yang, Wenyuan Martin boundary covers Floyd boundary (2017) (https://arxiv.org/abs/1708.02133)

[14] Gouëzel, Sébastien Martin boundary of random walks with unbounded jumps in hyperbolic groups, Ann. Probab., Volume 43 (2015), pp. 2374-2404 | DOI | MR | Zbl

[15] Hennequin, Paul-Louis Processus de Markoff en Cascade, Ann. Inst. Henri Poincaré, Volume 18 (1963), pp. 109-196 | Numdam | MR | Zbl

[16] Hruska, Geoffrey; Kleiner, Bruce Hadamard spaces with isolated flats, Geom. Topol., Volume 9 (2005), pp. 1501-1538 | DOI | MR | Zbl

[17] Hueber, Hermann; Müller, Detlef Asymptotics for some Green kernels on the Heisenberg group and the Martin boundary, Math. Ann., Volume 283 (1989), pp. 97-119 | DOI | MR | Zbl

[18] Ignatiouk-Robert, Irina Martin boundary of a killed random walk on a half-space, J. Theor. Probab., Volume 21 (2008), pp. 35-68 | DOI | MR | Zbl

[19] Ignatiouk-Robert, Irina; Loree, Christophe Martin boundary of a killed random walk on a quadrant, Ann. Probab., Volume 38 (2010), pp. 1106-1142 | DOI | MR | Zbl

[20] Kato, Tosio Perturbation Theory for Linear Operators, Springer, 1980 | Zbl

[21] Margulis, Gregori Positive harmonic functions on nilpotent groups, Sov. Math., Dokl., Volume 7 (1966), pp. 241-244 | MR

[22] Ney, Peter; Spitzer, Frank The Martin boundary for random walk, Trans. Am. Math. Soc., Volume 11 (1966), pp. 116-132 | MR | Zbl

[23] Pólya, Georg Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz, Math. Ann., Volume 84 (1921), pp. 149-160 | DOI | Zbl

[24] Raschel, Kilian; Lecouvey, Cédric t-Martin boundary of killed random walks in the quadrant, Séminaire de Probabilités (Lecture Notes in Mathematics), Volume 2168, Springer, 2016, pp. 305-323 | MR | Zbl

[25] Sawyer, Stanley Martin Boundaries and Random Walks, Harmonic functions on trees and buildings (Contemporary Mathematics), Volume 206, American Mathematical Society, 1997, pp. 17-44 | DOI | MR | Zbl

[26] Seneta, Eugene Non-negative Matrices and Markov Chains, Springer, 1981 | Zbl

[27] Spitzer, Frank Principles of Random Walks, Springer, 1976 | Zbl

[28] Woess, Wolfgang A description of the Martin boundary for nearest neighbour random walks on free products, Probability measures on groups VIII (Lecture Notes in Mathematics), Volume 1210, Springer, 1986, pp. 203-215 | DOI | MR | Zbl

[29] Woess, Wolfgang Random Walks on Infinite Graphs and Groups, Cambridge University Press, 2000 | Zbl

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