A dynamical-system picture of a simple branching-process phase transition

Abstract

This paper develops ideas from a previous paper described as `an appetizer for non-linear Wiener--Hopf theory', but is completely independent of that paper. It again considers only the simplest possible case in which the underlying motion of the branching particles is described by a two-state Markov chain. Key generating functions provide solutions of a simple two-dimensional dynamical system, and the main interest is in the way in which Probability Theory and ODE theory complement each other. ODE pictures convey rather strikingly a simple phase transition. No knowledge of either ODE theory or Wiener--Hopf theory is assumed.

18 pages. In: Bingham, N. H., and Goldie, C. M. (eds), Probability and Mathematical Genetics: Papers in Honour of Sir John Kingman. London Math. Soc. Lecture Note Series vol. 378. Cambridge: Cambridge Univ. Press, 2010