General branching processes as Markov fields

Abstract

The natural Markov structure for population growth is that of genetics: newborns inherit types from their mothers, and given those they are independent of the history of their earlier ancestry. This leads to Markov fields on the space of sets of individuals, partially ordered by descent. The structure of such fields is investigated. It is proved that this Markov property implies branching, i.e. the conditional independence of disjoint daughter populations. The process also has the strong Markov property at certain optional sets of individuals. An intrinsic martingale (indexed by sets of individuals) is exhibited, that catches the stochastic element of population development. The deterministic part is analyzed by Markov renewal methods. Finally the strong Markov property found is used to divide the population into conditionally independent subpopulations. On those classical limit theory for sums of independent random variables can be used to catch the asymptotic population development, as real time passes.