Fluctuations of a two-level critical branching system

Summary

Abstract. The critical dimension for the long-time behaviour of the highdensity fluctuation limit of a system of critical branching super Brownian motions is equal to 4, the same as the critical dimension for a related two–level superprocess.

Introduction.

Multilevel branching particle systems were introduced recently by Dawson and Hochberg [DH] as models for a class of hierarchically structured populations. In such a model individual particles migrate and branch, and in addition collections of particles are subject to independent branching mechanisms at the different levels of the hierarchical organization. The analysis of these models is complicated by the fact that the higher level branching leads to the absence of independence in the particle dynamics. Several aspects of multilevel branching systems have been investigated by Dawson, Hochberg and Wu [DHW], Dawson, Hochberg and Vinogradov [DHV1, DHV2], Dawson and Wu [DW], Etheridge [E], Gorostiza, Hochberg and Wakolbinger [GHW], Hochberg [H], and Wu [W1,W2, W3,W4]. Some of these references, specially [DHV2], include examples of areas of application where hierarchical branching structures are present.

Wu [W1,W2] studied the long-time behaviour of a critical two–level superprocess constructed from a system of branching Brownian motions in R^d. This process is obtained in a similar way as the usual (one-level) superprocesses, i.e., as a limit under a high-density/short-life/small-mass rescaling, which in the two-level case is applied simultaneously at the two levels. He proved that this critical two-level superprocess goes to extinction, as time tends to infinity, in dimensions d ≤ 4, and from his results in [Wl] it follows that it does not become extinct in dimensions d > 4. Gorostiza, Hochberg and Wakolbinger [GHW] proved the stronger result that persistence holds in dimensions d > 4.