Asymptotic behaviors of subcritical branching killed Lévy processes

In this paper, we investigate the asymptotic behaviors of the survival probability and maximal displacement of a subcritical branching killed Lévy process $X$ in $\mathbb{R}$. Let $\zeta$ denote the extinction time, $M_t$ be the maximal position of all the particles alive at time $t$, and $M:={\sup }_{t\ge 0}{M}_t$ be the all-time maximum. Under the assumption that the offspring distribution satisfies the $L\log L$ condition and some conditions on the spatial motion, we find the decay rate of the survival probability ${\mathbb{P}}_x(\zeta >t)$ and the tail behavior of $M_t$ as $t\to \mathrm{\infty }$. As a consequence, we establish a Yaglom-type theorem. We also find the asymptotic behavior of ${\mathbb{P}}_x(M>y)$ as $y\to \mathrm{\infty }$.