Abstract
Consider a closed subgroup Γ of the automorphism group of a homogeneous treeT, and assume that Γ acts transitively on the vertex set. Suppose that μ is a probability measure on Γ which has continuous density with respect to Haar measure and whose support is compact open and generates Γ as a closed semigroup. It is shown that the Martin boundary of Γ with respect to the random walk with law μ coincides with the space of ends ofT. This extends known results for free groups and applies, for example, to the affine group over a non archimedean local field.
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References
Ancona, A.: Positive harmonic functions and hyperbolicity. Potential Theory, Surveys and Problems (J. Král et al., eds.), pp. 1–23. Lect. Notes Math.1344. Berlin, Heidelberg, New York: Springer. 1988.
Azencott, R., Cartier, P.: Martin boundaries of random walks on locally compact groups. Proc. 6th Berkeley Sympos. on Math. Statistics and Probability3, pp. 87–129. Berkeley: Univ. of California Press. 1972.
Birkhoff, G.: Extensions of Jentzsch's theorem. Trans. Amer. Math. Soc.85, 219–227 (1957).
Cartier, P.: Fonctions harmoniques sur un arbre. Symposia Math.9, 203–270 (1972).
Cartwright, D. I., Kaimanovich, V. A., Woess, W.: Random walks on the affine group of local fields and homogeneous trees. Ann. Inst. Fourier (Grenoble)44, 1243–1288 (1994).
Cartwright, D. I., Soardi, P. M., Woess, W.: Martin and end compactifications of non locally finite graphs. Trans. Amer. Math. Soc.338, 670–693 (1993).
Derriennic, Y.: Sur la frontière de Martin des processus de Markov à temps discret. Ann. Inst. Henri Poincaré, Sect. B9, 233–258 (1973).
Derriennic, Y.: Marche aléatoire sur le groupe libre et frontière de Martin. Z. Wahrsch. Verw. Gebiete32, 261–276 (1975).
Doob, J. L.: Discrete potential theory and boundaries. J. Math. Mech.8, 433–458 (1959).
Élie, L.: Fonctions harmoniques positives sur le groupe affine. Probability Measures on Groups (H. Heyer, ed.), pp. 96–110, Lect. Notes Math.706. Berlin, Heidelberg, New York: Springer. 1979.
Figá-Talamanca, A., Nebbia, C.: Harmonic Analysis and Representation Theory for Groups Acting on Homogeneous Trees. Cambridge: Univ. Press. 1991.
Guivarc'h, Y., Keane, M., Roynette, B.: Marches Aléatoires sur les Groupes de Lie. Lect. Notes Math.624. Berlin, Heidelberg, New York: Springer. 1977.
Hewitt, E., Ross, K. A.: Abstract Harmonic Analysis I. Berlin, Heidelberg, New York: Springer. 1963.
Hunt, G. A.: Markoff chains and Martin boundaries. Illinois J. Math.4, 313–340 (1960).
Martin, R. S.: Minimal harmonic functions. Trans. Amer. Math. Soc.49, 137–172 (1941).
Ney, P., Spitzer, F.: The Martin boundary for random walk. Trans. Amer. Math. Soc.121, 116–132 (1966).
Picardello, M. A., Woess, W.: Martin boundaries of random walks: ends of trees and groups. Trans. Amer. Math. Soc.302, 185–205 (1987).
Revuz, D.: Markov Chains. Amsterdam: North Holland. 1975.
Serre, J.-P.: Local Fields. New York: Springer. 1979.
Serre, J.-P.: Trees. New York: Springer. 1980.
Soardi, P. M., Woess, W.: Amenability, unimodularity, and the spectral radius of random walks on infinite graphs. Math. Z.205, 471–486 (1990).
Woess, W.: Random walks on infinite graphs and groups: a survey on selected topics. Bull. London Math. Soc.26, 1–60 (1994).
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Woess, W. The martin boundary for harmonic functions on groups of automorphisms of a homogeneous tree. Monatshefte für Mathematik 120, 55–72 (1995). https://doi.org/10.1007/BF01470065
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DOI: https://doi.org/10.1007/BF01470065