Tails of extinction time and maximal displacement for critical branching killed L\'evy process

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