The Martin boundary of a free product of abelian groups

Dussaule, Matthieu

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 $\Gamma$, on définit son bord de Martin qui donne une manière de compactifier $\Gamma$ 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 $\mu$ on a finitely generated group $\Gamma$, its Martin boundary is a way to compactify $\Gamma$ 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.

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

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