Vladimir Alekseevich Vatutin

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.

A Decomposable Branching Process in a Markovian Environment

A population has two types of individuals, with each occupying an island. One of those, where individuals of type 1 live, offers a variable environment. Type 2 individuals dwell on the other island, in a constant environment. Only one-way migration is possible. We study then asymptotics of the survival probability in critical and subcritical cases.