Limit Theorems for Supercritical Branching Random Fields

Abstract

An infinite particle system in Rd is considered where the initial distribution is POISSONian and each initial particle gives rise to a supercritical age-dependent branching process with the particles moving randomly in space. Our approach differs from the usual: instead of the point measures determined by the locations of the particles at each time, we take the particles at a “final time” and observe the past histories of their ancestry lines. A law of large numbers and a central limit theorem are proved under a space-time scaling representing high density of particles and small mean particle lifetime. The fluctuation limit is a generalized GAUSS-MARKOV process with continuous trajectories and satisfies a deterministic evolution equation with generalized random initial condition. A more precise form of the central limit theorem is obtained in the case of particles performing BROWNian motion and having exponentially distributed lifetime.