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Branching Random Walks in Space–Time Random Environment: Survival Probability, Global and Local Growth Rates

Author

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  • Francis Comets

    (Université Paris Diderot—Paris 7)

  • Nobuo Yoshida

    (Kyoto University)

Abstract

We study the survival probability and the growth rate for branching random walks in random environment (BRWRE). The particles perform simple symmetric random walks on the d-dimensional integer lattice, while at each time unit, they split into independent copies according to time–space i.i.d. offspring distributions. The BRWRE is naturally associated with the directed polymers in random environment (DPRE), for which the quantity called the free energy is well studied. We discuss the survival probability (both global and local) for BRWRE and give a criterion for its positivity in terms of the free energy of the associated DPRE. We also show that the global growth rate for the number of particles in BRWRE is given by the free energy of the associated DPRE, though the local growth rate is given by the directional free energy.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:3:d:10.1007_s10959-009-0267-x
    DOI: 10.1007/s10959-009-0267-x
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    References listed on IDEAS

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    1. Hu, Yueyun & Yoshida, Nobuo, 2009. "Localization for branching random walks in random environment," Stochastic Processes and their Applications, Elsevier, vol. 119(5), pages 1632-1651, May.
    2. Liu, Quansheng & Watbled, Frédérique, 2009. "Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3101-3132, October.
    3. Carmona, Philippe & Hu, Yueyun, 2004. "Fluctuation exponents and large deviations for directed polymers in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 285-308, August.
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    Cited by:

    1. 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.
    2. 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.
    3. Xiaoqiang Wang & Chunmao Huang, 2017. "Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time," Journal of Theoretical Probability, Springer, vol. 30(3), pages 961-995, September.

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