Hierarchical Equilibria of Branching Populations

Donald A. Dawson (Carleton University)
Luis G. Gorostiza (Centro de Investigacion y de Estudios Avanzados, Mexico D.F., Mexico)
Anton Wakolbinger (Goethe Universitat, Frankfurt am Main, Germany)


Abstract. The objective of this paper is the study of the equilibrium behavior of a population on the hierarchical group $\Omega_N$ consisting of families of individuals undergoing critical branching random walk and in addition these families also develop according to a critical branching process. Strong transience of the random walk guarantees existence of an equilibrium for this two-level branching system. In the limit $N\to\infty$ (called the hierarchical mean field limit), the equilibrium aggregated populations in a nested sequence of balls $B^{(N)}_\ell$ of hierarchical radius $\ell$ converge to a backward Markov chain on $\mathbb{R_+}$. This limiting Markov chain can be explicitly represented in terms of a cascade of subordinators which in turn makes possible a description of the genealogy of the population.

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Pages: 316-381

Publication Date: April 26, 2004

DOI: 10.1214/EJP.v9-200


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