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One-dimensional branching random walks in a Markovian random environment

Part of: Equilibrium statistical mechanics Markov processes

Published online by Cambridge University Press:  14 July 2016

F. P. Machado  and
S. Yu. Popov
F. P. Machado*
Affiliation:
University of São Paulo
S. Yu. Popov*
Affiliation:
Institute for Problems of Information Transmission, Moscow
*
∗ Postal address: Department of Statistics, Institute of Mathematics and Statistics, University of São Paulo, Rua de Matão 1010, CEP 05508-900, São Paulo, SP, Brazil.
∗ Postal address: Department of Statistics, Institute of Mathematics and Statistics, University of São Paulo, Rua de Matão 1010, CEP 05508-900, São Paulo, SP, Brazil.
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Abstract

We study a one-dimensional supercritical branching random walk in a non-i.i.d. random environment, which considers both the branching mechanism and the step transition. This random environment is constructed using a recurrent Markov chain on a finite or countable state space. Criteria of (strong) recurrence and transience are presented for this model.

Keywords

Branching random walkrandom environmentstrong transiencestrong recurrence

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 82B41: Random walks, random surfaces, lattice animals, etc.
Type
Short Communications
Information
Journal of Applied Probability , Volume 37 , Issue 4 , December 2000 , pp. 1157 - 1163
Copyright
Copyright © by the Applied Probability Trust 2000 

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References

[1] Baillon, J.-B., Clément, Ph., Greven, A., and den Hollander, F. (1993). A variational approach to branching random walk in random environment. Ann. Prob. 21, 290–317.CrossRefGoogle Scholar
[2] Comets, F., Menshikov, M. V. and Popov, S. Yu. (1998). One-dimensional branching random walk in random environment: a classification. Markov Proc. Rel. Fields 4, 465–477.Google Scholar
[3] Comets, F., Menshikov, M. V. and Popov, S. Yu. (1998). Lyapunov functions for random walks and strings in random environment. Ann. Prob. 26, 1433–1445.CrossRefGoogle Scholar
[4] Den Hollander, F., Menshikov, M. V. and Popov, S. Yu. (1999). A note on transience versus recurrence for branching random walk in random environment. J. Statist. Phys. 95, 587–614.CrossRefGoogle Scholar
[5] Fayolle, G., Malyshev, V. A., and Menshikov, M. V. (1995). Topics in the Constructive Theory of Countable Markov Chains. Cambridge University Press.CrossRefGoogle Scholar
[6] Fleischmann, K., and Greven, A. (1992). Localization and selection in a mean field branching random walk in a random environment. Ann. Prob. 20, 2141–2163.CrossRefGoogle Scholar
[7] Greven, A., and den Hollander, F. (1992). Branching random walk in random environment: phase transition for local and global growth rates. Prob. Theor. Rel. Fields 91, 195–249.CrossRefGoogle Scholar
[8] Karpelevich, F. I., Kel'bert, M. Ya. and Suhov, Yu. M. (1994). The boundedness of branching Markov processes. In The Dynkin Festschrift. Markov Processes and their Applications, ed. Freidlin, M. Birkhauser, Boston, pp. 143–153.CrossRefGoogle Scholar
[9] Karpelevich, F. I. and Suhov, Yu. M. (1996). A criterion of boundedness of discrete branching random walk. In Classical and Modern Branching Processes, eds Atreya, K. and Jagers, P. Springer, New York, pp. 141–156.Google Scholar
[10] Karpelevich, F. I. and Suhov, Yu. M. (1997). Boundedness of one-dimensional branching Markov processes. J. Appl. Math. Stoch. Anal. 10, 307–332.CrossRefGoogle Scholar
[11] Menshikov, M. V., and Volkov, S. E. (1997). Branching Markov chains: qualitative characteristics. Markov Proc. Rel. Fields 3, 225–241.Google Scholar
[12] Révész, P. (1998). Supercritical branching random walk in d-dimensional random environment. In Applied Statistical Science III, eds Ahmed, S. E., Ahsanullah, M. and Sinha, B. K. Nova Science Publishers, Commack, NY, pp. 41–51.Google Scholar