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

Abstract

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