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The Extinction Rate of a Branching Random Walk with a Barrier in a Time-Inhomogeneous Random Environment

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  • You Lv

    (Donghua University)

Abstract

Consider a supercritical branching random walk in a time-inhomogeneous random environment. We impose a selection (called barrier) on survival in the following way. The position of the barrier may depend on the generation and the environment. In each generation, only the individuals born below the barrier can survive and reproduce. When the barrier causes the extinction of the system, we give the extinction rate in the sense of $$L^p~(p\ge 1)$$ L p ( p ≥ 1 ) . Moreover, we show the $$L^p$$ L p convergence of the small deviation probability for a random walk with random environment in time.

Suggested Citation

  • You Lv, 2025. "The Extinction Rate of a Branching Random Walk with a Barrier in a Time-Inhomogeneous Random Environment," Journal of Theoretical Probability, Springer, vol. 38(3), pages 1-39, September.
    • Handle: RePEc:spr:jotpro:v:38:y:2025:i:3:d:10.1007_s10959-025-01418-9
    DOI: 10.1007/s10959-025-01418-9
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    References listed on IDEAS

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    1. Gao, Zhiqiang & Liu, Quansheng, 2016. "Exact convergence rates in central limit theorems for a branching random walk with a random environment in time," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2634-2664.
    2. Mallein, Bastien, 2015. "Maximal displacement of a branching random walk in time-inhomogeneous environment," Stochastic Processes and their Applications, Elsevier, vol. 125(10), pages 3958-4019.
    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.
    4. Mallein, Bastien & Miłoś, Piotr, 2019. "Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3239-3260.
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