Asymptotic Behavior of the Two Level Measure Branching Process

Abstract

In this paper we consider a multilevel branching diffusion particle system and its diffusion approximation, which can be characterized as an $M(M(R^d))$-valued process. The long term behavior of the limiting process is studied. The main results are that if $d \leq 4$, then the two level $M(M(R^d))$-valued process suffers local extinction, and if $d = 4$, then the process has a self-similarity property.