Branching processes

Abstract

Branching processes, also called Galton–Watson processes after the name of their inventors (Francis Galton and Henry William Walton), are some of the simplest stochastic processes describing single-species dynamics. Such processes ignore the presence of a spatial structure but include general offspring distributions. The first reference to these processes is [39].

The first section gives a precise description of the model. The main assumption is that the number of offspring produced by each individual is independent and identically distributed across individuals and generations, which implies that the number of individuals evolves according to a discrete-time Markov chain.

The second section exhibits some useful connections with martingales suggesting that the process experiences a phase transition when the expected value μ of the offspring distribution is equal to one. From these connections, we prove exponentially fast extinction when μ < 1 and, more interestingly, almost sure extinction in the critical case μ = 1 using the martingale convergence theorem.