Explosion of Crump-Mode-Jagers processes with critical immediate offspring

TL;DR

The paper tackles explosion in general Crump-Mode-Jagers (CMJ) branching processes in the critical regime $\mathbf{E}[\xi\{0\}]=1$, where explosion depends on near-zero behavior of the reproduction point process $\xi$. It develops two general sufficient explosion criteria by comparing CMJ processes with Galton-Watson processes in varying environments and with Bellman-Harris processes, and it establishes a sharp integral criterion for Poisson reproduction when the cumulative mass near zero is convex. Central to the analysis is the smoothing transform $\mathcal{T}$ and its fixed points; the authors show that the survival distribution $\overline{F}$ of the explosion time is the minimal, unique (up to a shift) attracting fixed point, obtained as the limit of iterates $\mathcal{T}^n\phi$ for suitable $\phi$, via a multiplicative martingale argument. The results unify and extend classical explosion criteria for CMJ and Bellman-Harris processes, provide robust comparison tools, and yield a precise solvability criterion for the Poisson-reproduction case with practical implications for models where explosion corresponds to rapid, dense growth within finite time.

Abstract

We study the phenomenon of explosion in general (Crump-Mode-Jagers) branching processes, which refers to the event where an infinite number of individuals are born in finite time. In a critical setting where the expected number of immediate offspring per individual is exactly one, whether or not explosion occurs depends on the fine properties of the reproduction point process. We provide two sufficient conditions for explosion in these CMJ processes. The first uses a comparison with Galton-Watson processes in varying environments, while the second relies on a comparison with Bellman-Harris branching processes. Our main result is an equivalent characterization of explosion, expressed as an integral test, in the case where the reproduction point process is Poisson. For the derivation, we also study the fixed-point equation associated with a smoothing transform, which is known to describe the distribution of the explosion time. We use multiplicative martingales to show that this distribution is an attractive fixed point of the smoothing transform, which in particular implies its uniqueness modulo an additive shift.

Theorems & Definitions (48)

• Definition 2.1
• Proposition 2.2
• Remark 2.3
• Theorem 2.4
• Remark 2.5
• Theorem 2.6
• Theorem 2.7
• Remark 2.8
• Theorem 2.9
• Proposition 3.1