The objective of this paper is the study of the equilibrium behavior of a population on the hierarchical group ΩN 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 → ∞ (called the hierarchical mean field limit), the equilibrium aggregated populations in a nested sequence of balls Bℓ (N) of hierarchical radius ℓ converge to a backward Markov chain on 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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Electronic Journal of Probability
School of Mathematics and Statistics

Dawson, D.A, Gorostiza, L.G., & Wakolbinger, A. (2004). Hierarchical equilibria of branching populations. Electronic Journal of Probability, 9, 316–381.