Electronic Communications in Probability

Recurrence for branching Markov chains

Sebastian Müller

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The question of recurrence and transience of branching Markov chains is more subtle than for ordinary Markov chains; they can be classified in transience, weak recurrence, and strong recurrence. We review criteria for transience and weak recurrence and give several new conditions for weak recurrence and strong recurrence. These conditions make a unified treatment of known and new examples possible and provide enough information to distinguish between weak and strong recurrence. This represents a step towards a general classification of branching Markov chains. In particular, we show that in homogeneous cases weak recurrence and strong recurrence coincide. Furthermore, we discuss the generalization of positive and null recurrence to branching Markov chains and show that branching random walks on $Z$ are either transient or positive recurrent.

Article information

Electron. Commun. Probab., Volume 13 (2008), paper no. 54, 576-605.

Accepted: 24 November 2008
First available in Project Euclid: 6 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

spectral radius branching Markov chains recurrence transience strong recurrence positive recurrence

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Müller, Sebastian. Recurrence for branching Markov chains. Electron. Commun. Probab. 13 (2008), paper no. 54, 576--605. doi:10.1214/ECP.v13-1424. https://projecteuclid.org/euclid.ecp/1465233481

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  • Athreya, K.B.; Ney, P. E. Branching processes. Die Grundlehren der mathematischen Wissenschaften, Band 196. Springer-Verlag, New York-Heidelberg, 1972. xi+287 pp.
  • Benjamini, I.; Peres, Y. Markov chains indexed by trees. Ann. Probab. 22 (1994), no. 1, 219–243.
  • Benjamini, I.; Peres, Y. Tree-indexed random walks on groups and first passage percolation. Probab. Theory Related Fields 98 (1994), no. 1, 91–112.
  • Biggins, J. D. The first- and last-birth problems for a multitype age-dependent branching process. Advances in Appl. Probability 8 (1976), no. 3, 446–459.
  • Comets, F.; Menshikov, M. V.; Popov, S. Yu. One-dimensional branching random walk in a random environment: a classification. I Brazilian School in Probability (Rio de Janeiro, 1997). Markov Process. Related Fields 4 (1998), no. 4, 465–477.
  • Comets, Francis; Popov, Serguei. On multidimensional branching random walks in random environment. Ann. Probab. 35 (2007), no. 1, 68–114.
  • Dembo, A.; Zeitouni, O. Large deviations techniques and applications. Second edition. Applications of Mathematics (New York), 38. Springer-Verlag, New York, 1998. xvi+396 pp. ISBN: 0-387-98406-2
  • den Hollander, F.; Menshikov, M. V.; Popov, S. Yu. A note on transience versus recurrence for a branching random walk in random environment. J. Statist. Phys. 95 (1999), no. 3-4, 587–614.
  • Fayolle, G.; Malyshev, V. A.; Menshikov, M. V. Topics in the constructive theory of countable Markov chains. Cambridge University Press, Cambridge, 1995. iv+169 pp. ISBN: 0-521-46197-9
  • Gantert, N.; M¸ller, S. The critical branching Markov chain is transient. Markov Process. Related Fields. 12 (2006), no. 4, 805–814.
  • Hammersley, J. M. Postulates for subadditive processes. Ann. Probability 2 (1974), 652–680.
  • Hueter, I.; Lalley, Steven P. Anisotropic branching random walks on homogeneous trees. Probab. Theory Related Fields 116 (2000), no. 1, 57–88.
  • Kingman, J. F. C. The first birth problem for an age-dependent branching process. Ann. Probability 3 (1975), no. 5, 790–801.
  • Lyons, R. Random walks and percolation on trees. Ann. Probab. 18 (1990), no. 3, 931–958.
  • R. Lyons, with Y. Peres. Probability on Trees and Networks. Cambridge University Press. In preparation. Current version available at http://mypage.iu.edu/~rdlyons
  • Machado, F. P.; Popov, S. Yu. One-dimensional branching random walks in a Markovian random environment. J. Appl. Probab. 37 (2000), no. 4, 1157–1163.
  • Machado, F. P.; Popov, S. Yu. Branching random walk in random environment on trees. Stochastic Process. Appl. 106 (2003), no. 1, 95–106.
  • Menshikov, M. V.; Volkov, S. E. Branching Markov chains: qualitative characteristics. Markov Process. Related Fields 3 (1997), no. 2, 225–241.
  • Menshikov, M.; Petritis, D.; Volkov, S. Random environment on coloured trees. Bernoulli 13 (2007), no. 4, 966–980.
  • M¸ller, S. Recurrence and transience for branching random walks in an iid random environment. Markov Process. Related Fields. 14 (2008), no. 1, 115–130.
  • M¸ller, S. A criterion for transience of multidimensional branching random walk in random environment. Electr. J. of Probab. 13 (2008), 1189–1202.
  • Nagnibeda, T.; Woess, W. Random walks on trees with finitely many cone types. J. Theoret. Probab. 15 (2002), no. 2, 383–422.
  • Pemantle, R.; Stacey, A. M. The branching random walk and contact process on Galton-Watson and nonhomogeneous trees. Ann. Probab. 29 (2001), no. 4, 1563–1590.
  • Peres, Y. Probability on trees: an introductory climb. Lectures on probability theory and statistics (Saint-Flour, 1997), 193–280, Lecture Notes in Math., 1717, Springer, Berlin, 1999.
  • Schinazi, R. On multiple phase transitions for branching Markov chains. J. Statist. Phys. 71 (1993), no. 3-4, 507–511.
  • Stacey, A. Branching random walks on quasi-transitive graphs. Combinatorics, probability and computing (Oberwolfach, 2001). Combin. Probab. Comput. 12 (2003), no. 3, 345–358.
  • Vere-Jones, D. Ergodic properties of nonnegative matrices. I. Pacific J. Math. 22 1967 361–386.
  • Volkov, S. Branching random walk in random environment: fully quenched case. Markov Process. Related Fields 7 (2001), no. 2, 349–353.
  • Woess, W. Random walks on infinite graphs and groups. Cambridge Tracts in Mathematics, 138. Cambridge University Press, Cambridge, 2000. xii+334 pp. ISBN: 0-521-55292-3
  • Woess, W. Denumerable Markov Chains. To appear.