Asymptotic behavior for the quenched survival probability of a supercritical branching random walk in random environment with a barrier

2024/06/21 by You Lv, Lv, You, Wenming Hong +1
Mathematics · Physics and Astronomy · #60J80 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2406.15150

openalex publication_date 2024/06/21 · openalex created_date 2024/06/25 · openalex updated_date 2026/07/28

Abstract

We introduce a random barrier to a supercritical branching random walk in an i.i.d. random environment \L_n\ indexed by time n, i.e., in each generation, only the individuals born below the barrier can survive and reproduce. At generation n (n∈ℕ), the barrier is set as χ_n+ε n, where \χ_n\ is a random walk determined by the random environment. Lv & Hong (2024) showed that for almost every L:=\L_n\, the quenched survival probability (denoted by \varrho_L(ε)) of the particles system will be 0 (resp., positive) when ε≤ 0 (resp., ε>0). In the present paper, we prove that √(ε)log\varrho_L(ε) will converge in Probability/ almost surely/ in L^p to an explicit negative constant (depending on the environment) as ε\downarrow 0 under some integrability conditions respectively. This result extends the scope of the result of Gantert et al. (2011) to the random environment case.

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