Hierarchical equilibria of branching populations

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 B (N) ` of hierarchical radius ` 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. Research supported by NSERC (Canada) and a Max Planck Award for International Cooperation. 2 Research supported by CONACYT grant 37130-E (Mexico). 3 Research supported by DFG (SPP 1033) (Germany).