Abstract
We give an introductory survey on concepts and results concerning harmonic functions on infinite graphs with the goal of describing the interplay between graph structure and potential theory. A particular emphasis is on the connection between the Martin boundary for harmonic functions and the space of ends of the underlying graph. A variety of results is described.
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References
R. Azencott and P. Cartier: Martin boundaries of random walks on locally compact groups. Proc. 6th Berkeley Symposium on Math. Statistics and Probability3 (1972), 87 – 129.
J. R. Baxter: Restricted mean values and harmonic functions. Trans. Amer. Math. Soc.167 (1972), 451 – 463.
J. R. Baxter: Harmonic functions and mass cancellation. Trans. Amer. Math. Soc.245 (1978), 375 – 384.
P. Cartier: Fonction harmoniques sur un arbre. Symposia Math. 9 (1972), 203 – 270.
G. Choquet and J. Deny: Sur l’équation de convolution µ= µ* σ. C.R. Acad. Sci. Paris250 (1960), 799 – 801.
Y. Derriennic: Marche aléatoire sur le groupe libre et frontière de Martin. Zeitschr. Wahrscheinlichkeitstheorie Verw. Geb.32 (1975), 261 – 276.
J. L. Doob: Discrete potential theory and boundaries. J. Math. Mech8 (1959), 433 – 458.
J. L. Doob, J. L. Snell and R. E. Williamson: Application of boundary theory to sums of independent random variables. Contributions to Probability and Statistics, Stanford Univ. Press (1960), 182 – 197.
P. Doyle and J. L. Snell: Random Walks and Electric Networks. Carus Math. Monographs, Math. Assoc. Amer., 1984.
E. B. Dynkin and M. B. Malyutov: Random walks on groups with a finite number of generators. Soviet Math. Doklady2 (1961), 399 – 402.
W. Feller: Boundaries induced by nonnegative matrics. Trans. Amer. Math. Soc.83 (1956), 19 – 54.
H. Freudenthal: Über die Enden diskreter Räume und Gruppen. Comment. Math. Helv.17 (1944), 1 – 38.
H. Furstenberg: Random walks and discrete subgroups of Lie groups. Advances in Probability and Related Topics, Vol. 1 (P. Ney, ed.), Dekker, New York (1971), 1 – 63.
R. Halin: Über unendliche Wege in Graphen. Math. Annalen157 (1964), 125 – 137.
D. Heath: Functions possessing restricted mean value properties. Proc. Amer. Math. Soc.41 (1973), 588 – 595.
P. L. Hennequin: Processus de Markoff en cascade. Ann. Inst. H. Poincaré18 (1963), 109 – 196.
H. A. Jung: Connectivity in infinite graphs. Studies in Pure Math. (L. Mirsky, ed.), Academic Press, New York (1971), 137 – 143.
J. G. Kemeny, J. L. Snell and A. W. Knapp: Denumerable Markov Chains. 2nd ed., Springer, New York — Heidelberg — Berlin, 1976.
V. A. Kaimanovich and A. M. Vershik: Random walks on discrete groups: boundary and entropy. Annals Probab. 11 (1983), 457 – 490.
T. Lyons: A simple criterion for transience of a reversible Markov chain. Annals Probab. 11 (1983), 393 – 402.
W. Magnus: Noneuclidean Tesselations and their Groups. Academic Press, New York, 1974.
C. St. J. A. Nash-Williams: Random walks and electric currents in networks. Proc. Cambridge Phil. Soc.55 (1959), 181 – 194.
P. Ney and F. Spitzer: The Martin boundary for random walk. Trans. Amer. Math. Soc.121 (1966), 116 – 132.
R. R. Phelps: Lectures on Choquet’s Theorem. Van Nostrand Math. Studies, Vol. 7, New York, 1966.
M. A. Picardello and W. Woess: Martin boundaries of random walks: ends of trees and groups. Trans. Amer. Math. Soc.302 (1987), 185 – 205.
M. A. Picardello and W. Woess: Harmonic functions and ends of graphs. Proc. Edinburgh Math. Soc.31 (1988), 457 – 461.
M. A. Picardello and W. Woess: A converse to the mean value property on homogeneous trees. Trans. Amer. Math. Soc in print
M. A. Picardello and P. Sjögren: Preprint, Univ. Maryland (1988).
N. Polat: Aspects topologiques de la séparation dans les graphes infinis. Math. Zeitschr.165 (1979), 73 – 100.
C. Series: Martin boundaries of random walks on Fuchsian groups. Israel J. Math.44 (1983), 221 – 242.
F. Spitzer: Principles of Random Walk. Van Nostrand, Princeton, 1974.
J. C. Taylor: The Martin boundaries of equivalent sheaves. Ann. Inst. Fourier20 (1970), 433 – 456.
C. Thomassen: Resistances and currents in infinite electrical networks. J. Combin. Theory Ser. B in print.
W. Veech: A zero-one law for a class of random walks and a converse to Gauss’ mean value theorem. Annals Math. 97 (1973), 189 – 216.
W. Veech: A converse to the mean value theorem for harmonic functions. Amer. J. Math97 (1975), 1007 – 1027.
W. Woess: A description of the Martin boundary for nearest neighbour random walks on free products. “Probability Measures on Groups”, Springer Led. Notes in Math.1210 (1986), 203 – 215.
W. Woess: Harmonic functions on infinite graphs. Rendiconti Sera. Mat. Fis. Milano56 (1986), 53 – 63.
W. Woess: Graphs and groups with tree-like properties. J. Combin. Theory Ser. B in print.
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Picardello, M.A., Woess, W. (1990). Ends of Infinite Graphs, Potential Theory and Electrical Networks. In: Hahn, G., Sabidussi, G., Woodrow, R.E. (eds) Cycles and Rays. NATO ASI Series, vol 301. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0517-7_15
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DOI: https://doi.org/10.1007/978-94-009-0517-7_15
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