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Harmonic functions and ends of graphs

Published online by Cambridge University Press:  13 July 2011

Massimo A. Picardello  and
Wolfgang Woess
Massimo A. Picardello
Affiliation:
Dipartimento di Matematica “Guido Castelnuovo”Università di RomaI-00100 Roma, Italy Institut für Mathematik und Angewandte GeometrieMontanuniversitätA-8700 LeobenAustria
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In this note we discuss extensions of results [5], where transient random walks are considered, whose transition matrix is compatible with a tree-structure of the underlying discrete state space. Notation is generally as in [5], with the exception that instead of a tree T we consider an arbitrary graph Γ, which is locally finite, connected, and whose vertex set S is infinite. The edge set E is unoriented, there are no multiple edges. We consider a transition operator P, given by a stochastic matrix (p(u, v))u, v ∈ S, which gives rise to a transient Markov chain (“random walk”), related to the graph structure by:

(i) if p(u, v)>0 then [u, v] ∈ E,

(ii) there is an M>0 such that p(k)(u, v)>0 for some k=k(u, v)≦M whenever [u, v]∈E (“uniform irreducibility”).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 31 , Issue 3 , October 1988 , pp. 457 - 461
Copyright
Copyright © Edinburgh Mathematical Society 1988

References

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