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Martin boundaries of cartesian products of markov chains

Published online by Cambridge University Press:  22 January 2016

Massimo A. Picardello  and
Wolfgang Woess
Massimo A. Picardello
Affiliation:
Dipartimento di Matematica, Universitá di Roma “Tor Vergata”, 00133 Roma, Italy
Wolfgang Woess
Affiliation:
Dipartimento di Matematica, Universitá di Milano, 20133 Milano, Italy
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Let P and Q be the stochastic transition operators of two time-homogeneous, irreducible Markov chains with countable, discrete state spaces X and Y, respectively. On the Cartesian product Z = X x Y, define a transition operator of the form Ra = a·P + (1 — a) · Q, 0 < a < 1, where P is considered to act on the first variable and Q on the second. The principal purpose of this paper is to describe the minimal Martin boundary of Ra (consisting of the minimal positive eigenfunctions of Ra with respect to some eigenvalue t, also called t-harmonic functions) in terms of the minimal Martin boundaries of P and Q.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 128 , December 1992 , pp. 153 - 169
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1992

References

[Do] Doob, J. L., Discrete potential theory and boundaries, J. Math. Meth., 8 (1959), 433–458.Google Scholar
[Fr] Freire, A., On the Martin boundary of Riemannian products, in J. Diff. Geom., 33 (1991), 215–232 Google Scholar
[GT] Guivarc’h, Y., Taylor, J. C., The Martin compactification of the polydisc at the bottom of the positive spectrum, Colloquium Math., 50/51 (1990), 537–546.Google Scholar
[Hu] Hunt, G. A., Markoff chains and Martin boundaries, Illinois J. Math., 4 (1960), 313–340.Google Scholar
[Ka] Kaimanovich, V. A., The differential entropy of the boundary of a random walk on a group, Russian Math. Surveys, 38:5 (1983), 142–143.Google Scholar
[KV] Kaimanovich, V. A., Vershik, A.M., Random walks on discrete groups: boundary and entropy, Ann. Prob., 11 (1983), 457–490.Google Scholar
[KSK] Kemeny, J. G., Snell, J. L., Knapp, A. W., Denumerable Markov Chains, 2nd ed., Springer, New York-Heidelberg-Berlin, 1976.Google Scholar
[M1] Molchanov, S. A., On the Martin boundaries for the direct products of Markov chains, Theory of Prob. and Its Appl., 12 (1967), 307–314.Google Scholar
[M2] Molchanov, S. A., Martin boundaries for the direct product of Markov processes, Siberian J. Math., 11 (1970), 280–287.Google Scholar
[PS] Picardello, M. A., Sjogren, P., Boundary behaviour of eigenfunctions of the Laplacian in a bi-tree, J. Reine Angew. Math., 424 (1992), 137–148.Google Scholar
[PW] Picardello, M. A., Woess, W., Examples of stable Martin boundaries of Markov chains, in: Potential Theory (ed. Kishi, M.), de Gruyter. Berlin-New York (1991), 261–270.Google Scholar
[Pr] Pruitt, W. E., Eigenvalues of nonnegative matrices, Ann. Math. Statistics., 35 (1964), 1797–1800.Google Scholar
[Ta] Taylor, J. C., The product of minimal functions is minimal, Bull. London Math. Soc. 22 (1990), 499–504, erratum, Bull, London Math. Soc. 24 (1991), 379–380.Google Scholar
[Ve] Vere-Jones, D., Geometric ergodicity in denumerable Markov chains, Quart. J. Math. Oxford, 13 (1962), 7–28.Google Scholar