Branching systems with long-living particles at the critical dimension

A spatial branching process is considered in which particles have a lifetime law with a tail index smaller than one. It is shown that at the critical dimension, unlike classical branching particle systems the population does not suffer local extinction when started from a spatially homogeneous Poissonian initial population. In fact, persistent convergence to a mixed Poissonian particle system is shown. The random intensity of the limiting process is characterized in law by the random density in a space point of a related age-dependent superprocess at a fixed time. The proof relies on a refined study of the system starting from asymptotically large but finite initial populations.