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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Harmonic functions on hyperbolic graphs


Author: Camille Petit
Journal: Proc. Amer. Math. Soc. 140 (2012), 235-248
MSC (2010): Primary 31C05, 05C81; Secondary 60J45, 60D05, 60J50, 20F67
Published electronically: May 17, 2011
MathSciNet review: 2833536
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Abstract: We consider admissible random walks on hyperbolic graphs. For a given harmonic function on such a graph, we prove that asymptotic properties of non-tangential boundedness and non-tangential convergence are almost everywhere equivalent. The proof is inspired by the works of F. Mouton in the cases of Riemannian manifolds of pinched negative curvature and infinite trees. It involves geometric and probabilitistic methods.


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Additional Information

Camille Petit
Affiliation: Institut Fourier UMR 5582 UJF-CNRS, Université Joseph Fourier Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint Martin d’Hères, France
Email: camille.petit@ujf-grenoble.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10931-6
PII: S 0002-9939(2011)10931-6
Keywords: Harmonic functions, hyperbolic graphs, random walks, boundary at infinity, Fatou’s theorem, non-tangential convergence
Received by editor(s): July 2, 2009
Received by editor(s) in revised form: August 7, 2010, and November 10, 2010
Published electronically: May 17, 2011
Communicated by: Mario Bonk
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.