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Asymptotic behaviors of subcritical branching killed Brownian motion with drift
- Publication Year :
- 2024
-
Abstract
- In this paper, we study asymptotic behaviors of a subcritical branching killed Brownian motion with drift $-\rho$ and offspring distribution $\{p_k:k\ge 0\}$. Let $\widetilde{\zeta}^{-\rho}$ be the extinction time of this subcritical branching killed Brownian motion, $\widetilde{M}_t^{-\rho}$ the maximal position of all the particles alive at time $t$ and $\widetilde{M}^{-\rho}:=\max_{t\ge 0}\widetilde{M}_t^{-\rho}$ the all time maximal position. Let $\mathbb{P}_x$ be the law of this subcritical branching killed Brownian motion when the initial particle is located at $x\in (0,\infty)$. Under the assumption $\sum_{k=1}^\infty k (\log k) p_k <\infty$, we establish the decay rates of $\mathbb{P}_x(\widetilde{\zeta}^{-\rho}>t)$ and $\mathbb{P}_x(\widetilde{M}^{-\rho}>y)$ as $t$ and $y$ tend to $\infty$ respectively. We also establish the decay rate of $\mathbb{P}_x(\widetilde{M}_t^{-\rho}>z(t,\rho))$ as $t\to\infty$, where $z(t,\rho)=\sqrt{t}z-\rho t$ for $\rho\leq 0$ and $z(t,\rho)=z$ for $\rho>0$. As a consequence, we obtain a Yaglom-type limit theorem.
- Subjects :
- Mathematics - Probability
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2407.01816
- Document Type :
- Working Paper