Brownian motion on treebolic space: positive harmonic functions

Bendikov, Alexander; Saloff-Coste, Laurent; Salvatori, Maura; Woess, Wolfgang

Annales de l'Institut Fourier, Tome 66 (2016) no. 4, pp. 1691-1731.

Ce travail est dedié à une étude de la théorie du potentiel sur l’espace arbolique, i.e., le produit horcyclique d’un ârbre régulier avec le demi-plan hyperbolique supérieur. En se basant sur l’analyse sur les complexes à bandes Riemanniennes développée par les auteurs, on considère une famille de Laplaciens avec deux paramètres concernant la dérive verticale. On examine les fonctions harmoniques associées à ces Laplaciens.

This paper studies potential theory on treebolic space, that is, the horocyclic product of a regular tree and hyperbolic upper half plane. Relying on the analysis on strip complexes developed by the authors, a family of Laplacians with “vertical drift” parameters is considered. We investigate the positive harmonic functions associated with those Laplacians.

Classification : 31C05, 60J50, 53C23, 05C05
Keywords: Tree, hyperbolic plane, horocyclic product, quantum complex, Laplacian, positive harmonic functions
Mot clés : Arbre, plan hyperbolique, produit horocyclique, complexe quantique, Laplacien, fonctions harmoniques positives

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