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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

REFERENCES

1.Birkhoff, G., Extensions of Jentzsch's theorem, Trans. Amer. Math. Soc. 85 (1957), 219–227.Google Scholar
2.Freudenthal, H., Über die Enden diskreter Räume und Gruppen, Comment. Math. Helv. 17 (1944), 1–38.Google Scholar
3.Halin, R., Über unendliche Wege in Graphen, Math. Ann. 157 (1964), 125–137.Google Scholar
4.Jung, H. A., Connectivity in infinite graphs, Studies in Pure Math. (Ed. Mirsky, L., Academic Press, 1971), 137–143.Google Scholar
5.Picardello, M. A. and Woess, W., Martin boundaries of random walks: Ends of trees and groups, Trans. Amer. Math. Soc. 302 (1987), 185–205.Google Scholar
6.Seneta, E., Non-Negative Matrices and Markov Chains, 2nd ed. (Springer, 1981).CrossRefGoogle Scholar
7.Taylor, J. C., The Martin boundaries of equivalent sheaves, Ann. Inst. Fourier 20 (1970), 433–456.Google Scholar
8.Woess, W., Graphs and groups with tree-like properties, (1986), preprint.Google Scholar