Tails of extinction time and maximal displacement for critical branching killed Lévy process

In this paper, we study asymptotic behaviors of the tails of extinction time and maximal displacement of a critical branching killed Lévy process $(Z_t^{(0,\mathrm{\infty })}{)}_{t\ge 0}$ in $\mathbb{R}$, in which all particles (and their descendants) are killed upon exiting $(0,\mathrm{\infty })$. Let ${\zeta }^{(0,\mathrm{\infty })}$ and $M_t^{(0,\mathrm{\infty })}$ be the extinction time and maximal position of all the particles alive at time $t$ of this branching killed Lévy process and define $M^{(0,\mathrm{\infty })}:={\sup }_{t\ge 0}{M}_t^{(0,\mathrm{\infty })}$. Under the assumption that the offspring distribution belongs to the domain of attraction of an $\alpha$-stable distribution, $\alpha \in (1,2]$, and some moment conditions on the spatial motion, we give the decay rates of the survival probabilities $${\mathbb{P}}_y({\zeta }^{(0,\mathrm{\infty })}>t),{\mathbb{P}}_{\sqrt{t}y}({\zeta }^{(0,\mathrm{\infty })}>t)$$ and the tail probabilities $${\mathbb{P}}_y(M^{(0,\mathrm{\infty })}\ge x),{\mathbb{P}}_{xy}(M^{(0,\mathrm{\infty })}\ge x)\mathrm{.}$$ We also study the scaling limits of $M_t^{(0,\mathrm{\infty })}$ and the point process $Z_t^{(0,\mathrm{\infty })}$ under ${\mathbb{P}}_{\sqrt{t}y}(\cdot \mathrm{\mid }{\zeta }^{(0,\mathrm{\infty })}>t)$ and ${\mathbb{P}}_y(\cdot \mathrm{\mid }{\zeta }^{(0,\mathrm{\infty })}>t)$. The scaling limits under ${\mathbb{P}}_{\sqrt{t}y}(\cdot \mathrm{\mid }{\zeta }^{(0,\mathrm{\infty })}>t)$ are represented in terms of super killed Brownian motion.