Speaker: 侯浩杰，北京大学         

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

Language: Chinese 

Time & Venue: 2024.05.31 14:00-14:40 南楼N620

Abstract: In this talk, 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 $\zeta^{(0,\infty)}$ and $M_t^{(0,\infty)}$ be the extinction time and maximal position of all the particles alive at time $t$ of this branching killed L\'{e}vy process anddefine $M^{(0,\infty)}: = \sup_{t\geq 0} M_t^{(0,\infty)}$. Under the assumption that the offspring distribution belongs to the domain of attraction of an $\alpha$-stable distribution, $\alpha\in (1, 2]$, and some moment conditions on the spatial motion, we give the decay rates of the survival probabilities $$\mathbb{P}_{y}(\zeta^{(0,\infty)}>t), \quad \mathbb{P}_{\sqrt{t}y} (\zeta^{(0,\infty)}>t)$$ and the tail probabilities $$\mathbb{P}_{y} (M^{(0,\infty)}\geqx), \quad \mathbb{P}_{xy} (M^{(0,\infty)}\geq x).$$ We also study the scaling limits of $M_t^{(0,\infty)}$ and the point process $Z_t^{(0,\infty)}$ under $\mathbb{P}_{\sqrt{t}y}(\cdot |\zeta^{(0,\infty)}>t)$ and $\mathbb{P}_y(\cdot |\zeta^{(0,\infty)}>t)$. The scaling limits under $\mathbb{P}_{\sqrt{t}y} (\cdot | \zeta^{(0,\infty)}>t)$ are represented in terms of  super killed Brownian motion. This is a joint work with Yan-Xia Ren and Renming Song.