Asymptotic behavior for the quenched survival probability of a supercritical branching random walk in random environment with a barrier
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909228927549440 |
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| author | Lv, You Hong, Wenming |
| author_facet | Lv, You Hong, Wenming |
| contents | We introduce a random barrier to a supercritical branching random walk in an i.i.d. random environment $\{\mathcal{L}_n\}$ indexed by time $n,$ i.e., in each generation, only the individuals born below the barrier can survive and reproduce. At generation $n$ ($n\in\mathbb{N}$), the barrier is set as $χ_n+\varepsilon n,$ where $\{χ_n\}$ is a random walk determined by the random environment. Lv \& Hong (2024) showed that for almost every $\mathcal{L}:=\{\mathcal{L}_n\},$ the quenched survival probability (denoted by $\varrho_{\mathcal{L}}(\varepsilon)$) of the particles system will be 0 (resp., positive) when $\varepsilon\leq 0$ (resp., $\varepsilon>0$). In the present paper, we prove that $\sqrt{\varepsilon}\log\varrho_\mathcal{L}(\varepsilon)$ will converge in Probability/ almost surely/ in $L^p$ to an explicit negative constant (depending on the environment) as $\varepsilon\downarrow 0$ under some integrability conditions respectively. This result extends the scope of the result of Gantert et al. (2011) to the random environment case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_15150 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic behavior for the quenched survival probability of a supercritical branching random walk in random environment with a barrier Lv, You Hong, Wenming Probability 60J80 We introduce a random barrier to a supercritical branching random walk in an i.i.d. random environment $\{\mathcal{L}_n\}$ indexed by time $n,$ i.e., in each generation, only the individuals born below the barrier can survive and reproduce. At generation $n$ ($n\in\mathbb{N}$), the barrier is set as $χ_n+\varepsilon n,$ where $\{χ_n\}$ is a random walk determined by the random environment. Lv \& Hong (2024) showed that for almost every $\mathcal{L}:=\{\mathcal{L}_n\},$ the quenched survival probability (denoted by $\varrho_{\mathcal{L}}(\varepsilon)$) of the particles system will be 0 (resp., positive) when $\varepsilon\leq 0$ (resp., $\varepsilon>0$). In the present paper, we prove that $\sqrt{\varepsilon}\log\varrho_\mathcal{L}(\varepsilon)$ will converge in Probability/ almost surely/ in $L^p$ to an explicit negative constant (depending on the environment) as $\varepsilon\downarrow 0$ under some integrability conditions respectively. This result extends the scope of the result of Gantert et al. (2011) to the random environment case. |
| title | Asymptotic behavior for the quenched survival probability of a supercritical branching random walk in random environment with a barrier |
| topic | Probability 60J80 |
| url | https://arxiv.org/abs/2406.15150 |