Asymptotic behaviors of subcritical branching killed Lévy processes
Yan-Xia Ren, Renming Song, Yaping Zhu
TL;DR
The paper characterizes sharp large-time asymptotics for a subcritical branching killed Lévy process on the real line, focusing on survival probabilities and maximal displacement. It blends Lévy process fluctuation theory, scale-function techniques, and conditioned limit theorems to derive precise decay rates that depend on the drift regime of the underlying motion, and it establishes a Yaglom-type limit for the process conditioned on survival. The results include explicit asymptotics for P_x(ζ>t) and for P_x(M_t>·) across drift regimes, and a universal exponential tail for the all-time maximum M in the spectrally negative setting, with recovery of prior Brownian-based findings as special cases. Together, these contributions provide a comprehensive asymptotic picture for subcritical branching killed Lévy processes and illuminate the role of killing and spatial motion in extreme-event behavior.
Abstract
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 $ζ$ 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 $\mathbb{P}_x(ζ>t)$ and the tail behavior of $M_t$ as $t\to\infty$. As a consequence, we establish a Yaglom-type theorem. We also find the asymptotic behavior of $\mathbb{P}_x(M>y)$ as $y\to\infty$.