Tails of extinction time and maximal displacement for critical branching killed L\'{e}vy process

In this paper, we study asymptotic behaviors of the tails of extinction time and maximal displacement of a critical branching killed L\'{e}vy process (Zₜ(0,∞))t≥ 0 in R, in which all particles (and their descendants) are killed upon exiting (0, ∞). Let ζ(0,∞) and Mₜ(0,∞) be the extinction time and maximal position of all the particles alive at time t of this branching killed L\'{e}vy process and define M(0,∞): = t≥ 0 Mₜ(0,∞). Under the assumption that the offspring distribution belongs to the domain of attraction of an α-stable distribution, α∈ (1, 2], and some moment conditions on the spatial motion, we give the decay rates of the survival probabilities Py(ζ(0,∞)>t), P√ty(ζ(0,∞)>t) and the tail probabilities Py(M(0,∞)≥ x), Pxy(M(0,∞)≥ x). We also study the scaling limits of Mₜ(0,∞) and the point process Zₜ(0,∞) under P√ty(· |ζ(0,∞)>t) and Py(· |ζ(0,∞)>t). The scaling limits under P√ty(· |ζ(0,∞)>t) are represented in terms of super killed Brownian motion.