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Comportement asymptotique des fonctions harmoniques sur les arbres

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Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 1729)

Abstract

Considering a random walk on an infinite tree, we suppose that transition probabilities are “reasonnable”, id est that they are bounded between two constants taken in (0, 1/2). It is shown that, for a given harmonic function on the tree, properties of radial convergence, radial boundedness, finiteness of radial energy and corresponding stochastic notions are all equivalent at almost each point of the geometric boundary. The idea of the proof comes from an analoguous result on Riemannian manifolds of pinched negative curvature due to the author.

Mots-clés

  • fonctions harmoniques
  • arbres
  • théorème de Fatou
  • marches aléatoires

Classification math.

  • 05C05
  • 31C20
  • 31C35
  • 60J15
  • 60J50

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Mouton, F. (2000). Comportement asymptotique des fonctions harmoniques sur les arbres. In: Azéma, J., Ledoux, M., Émery, M., Yor, M. (eds) Séminaire de Probabilités XXXIV. Lecture Notes in Mathematics, vol 1729. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0103813

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  • DOI: https://doi.org/10.1007/BFb0103813

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  • Print ISBN: 978-3-540-67314-9

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