Asymptotic behaviors of subcritical branching killed Brownian motion with drift

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 ${\tilde{\zeta }}^{-\rho }$ be the extinction time of this subcritical branching killed Brownian motion, ${\tilde{M}}_t^{-\rho }$ the maximal position of all the particles alive at time $t$ and ${\tilde{M}}^{-\rho }:={\max }_{t\ge 0}{\tilde{M}}_t^{-\rho }$ 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 ${\mathbb{P}}_x({\tilde{\zeta }}^{-\rho }>t)$ and ${\mathbb{P}}_x({\tilde{M}}^{-\rho }>y)$ as $t$ and $y$ tend to $\infty$ respectively. We also establish the decay rate of ${\mathbb{P}}_x({\tilde{M}}_t^{-\rho }>z(t,\rho ))$ as $t\to \infty$, where $z(t,\rho )=\sqrt{t}z-\rho t$ for $\rho \le 0$ and $z(t,\rho )=z$ for $\rho >0$. As a consequence, we obtain a Yaglom-type limit theorem.