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Survival of Branching Random Walks in Random Environment

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Abstract

We study survival of nearest-neighbor branching random walks in random environment (BRWRE) on ℤ. A priori there are three different regimes of survival: global survival, local survival, and strong local survival. We show that local and strong local survival regimes coincide for BRWRE and that they can be characterized with the spectral radius of the first moment matrix of the process. These results are generalizations of the classification of BRWRE in recurrent and transient regimes. Our main result is a characterization of global survival that is given in terms of Lyapunov exponents of an infinite product of i.i.d. 2×2 random matrices.

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Authors and Affiliations

  1. Fachbereich Mathematik und Informatik, CeNos Center for Nonlinear Science and Institut für Mathematische Statistik, Einsteinstrasse 62, 48149, Münster, Germany

    Nina Gantert

  2. Institut für Mathematische Strukturtheorie, Technische Universität Graz, Steyrergasse 30, 8010, Graz, Austria

    Sebastian Müller

  3. Department of Statistics, Institute of Mathematics, Statistics and Scientific Computation, University of Campinas—UNICAMP, P.O. Box 6065, CEP 13083-970, Campinas, SP, Brazil

    Serguei Popov

  4. Department of Statistics, Institute of Mathematics, Statistics and Scientific Computation, University of Campinas—UNICAMP, P.O. Box 6065, CEP 13083-970, Campinas, SP, Brazil

    Marina Vachkovskaia

Authors
  1. Nina Gantert
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  2. Sebastian Müller
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  3. Serguei Popov
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  4. Marina Vachkovskaia
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Correspondence to Serguei Popov.

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Gantert, N., Müller, S., Popov, S. et al. Survival of Branching Random Walks in Random Environment. J Theor Probab 23, 1002–1014 (2010). https://doi.org/10.1007/s10959-009-0227-5

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  • DOI: https://doi.org/10.1007/s10959-009-0227-5

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