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Asymptotic behaviors of subcritical branching killed Brownian motion with drift

In this paper, we study asymptotic behaviors of a subcritical branching killed Brownian motion with drift $-ρ$ and offspring distribution $\{p_k:k\ge 0\}$. Let $\widetildeζ^{-ρ}$ be the extinction time of this subcritical branching killed Brownian motion, $\widetilde{M}_t^{-ρ}$ the maximal position of all the particles alive at time $t$ and $\widetilde{M}^{-ρ}:=\max_{t\ge 0}\widetilde{M}_t^{-ρ}$ 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 $\mathbb{P}_x(\widetildeζ^{-ρ}>t)$ and $\mathbb{P}_x(\widetilde{M}^{-ρ}>y)$ as $t$ and $y$ tend to $\infty$ respectively. We also establish the decay rate of $\mathbb{P}_x(\widetilde{M}_t^{-ρ}>z(t,ρ))$ as $t\to\infty$, where $z(t,ρ)=\sqrt{t}z-ρt$ for $ρ\leq 0$ and $z(t,ρ)=z$ for $ρ>0$. As a consequence, we obtain a Yaglom-type limit theorem.