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Survival of Branching Random Walks in Random Environment

Author

Listed:
  • Nina Gantert

    (CeNos Center for Nonlinear Science and Institut für Mathematische Statistik)

  • Sebastian Müller

    (Technische Universität Graz)

  • Serguei Popov

    (University of Campinas—UNICAMP)

  • Marina Vachkovskaia

    (University of Campinas—UNICAMP)

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.

Suggested Citation

  • Nina Gantert & Sebastian Müller & Serguei Popov & Marina Vachkovskaia, 2010. "Survival of Branching Random Walks in Random Environment," Journal of Theoretical Probability, Springer, vol. 23(4), pages 1002-1014, December.
    • Handle: RePEc:spr:jotpro:v:23:y:2010:i:4:d:10.1007_s10959-009-0227-5
    DOI: 10.1007/s10959-009-0227-5
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    Citations

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    Cited by:

    1. Makoto Nakashima, 2013. "Minimal Position of Branching Random Walks in Random Environment," Journal of Theoretical Probability, Springer, vol. 26(4), pages 1181-1217, December.
    2. Onur Gün & Wolfgang König & Ozren Sekulović, 2015. "Moment Asymptotics for Multitype Branching Random Walks in Random Environment," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1726-1742, December.
    3. Vincent Bansaye, 2019. "Ancestral Lineages and Limit Theorems for Branching Markov Chains in Varying Environment," Journal of Theoretical Probability, Springer, vol. 32(1), pages 249-281, March.

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