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Moment Asymptotics for Multitype Branching Random Walks in Random Environment

Author

Listed:
  • Onur Gün

    (Weierstrass Institute Berlin)

  • Wolfgang König

    (Weierstrass Institute Berlin
    Institute for Mathematics, TU Berlin)

  • Ozren Sekulović

    (University of Montenegro)

Abstract

We study a discrete-time multitype branching random walk on a finite space with finite set of types. Particles move in space according to a Markov chain whereas offspring distributions are given by a random field that is fixed throughout the evolution of the particles. Our main interest lies in the averaged (annealed) expectation of the population size, and its long-time asymptotics. We first derive, for fixed time, a formula for the expected population size with fixed offspring distributions, which is reminiscent of a Feynman–Kac formula. We choose Weibull-type distributions with parameter $$1/\rho _{ij}$$ 1 / ρ i j for the upper tail of the mean number of $$j$$ j type particles produced by an $$i$$ i type particle. We derive the first two terms of the long-time asymptotics, which are written as two coupled variational formulas, and interpret them in terms of the typical behavior of the system.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:4:d:10.1007_s10959-014-0551-2
    DOI: 10.1007/s10959-014-0551-2
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    References listed on IDEAS

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    1. Machado, F. P. & Popov, S. Yu., 2003. "Branching random walk in random environment on trees," Stochastic Processes and their Applications, Elsevier, vol. 106(1), pages 95-106, July.
    2. Francis Comets & Nobuo Yoshida, 2011. "Branching Random Walks in Space–Time Random Environment: Survival Probability, Global and Local Growth Rates," Journal of Theoretical Probability, Springer, vol. 24(3), pages 657-687, September.
    3. Biggins, J. D. & Cohn, H. & Nerman, O., 1999. "Multi-type branching in varying environment," Stochastic Processes and their Applications, Elsevier, vol. 83(2), pages 357-400, October.
    4. 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.
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    Cited by:

    1. Vladimir Kutsenko & Stanislav Molchanov & Elena Yarovaya, 2024. "Branching Random Walks in a Random Killing Environment with a Single Reproduction Source," Mathematics, MDPI, vol. 12(4), pages 1-22, February.

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