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

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.