2003/10/15 by D. A. Dawson, Donald A. Dawson, Luis G. Gorostiza +6
Biochemistry, Genetics and Molecular Biology · Mathematics · #60G60 #60J60 #60J80 #FOS: Biological sciences #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Populations and Evolution (q-bio.PE) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #math.PR #msc:60G60 #msc:60J60 #msc:60J80 #q-bio.PE
paper · pdf · doi:10.48550/arxiv.math/0310229
62 pages
arxiv created 2003/10/15 · openalex publication_date 2003/10/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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)_ℓ of hierarchical radius ℓ converge to a backward Markov chain on \mathbbR+. 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.