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

TL;DR

This paper explores a simple branching process with two-state Markov motion, using dynamical systems and ODE theory to reveal a clear phase transition, bridging probability and differential equations without prior specialized knowledge.

Contribution

It introduces a dynamical systems approach to analyze phase transitions in a simple branching process with Markovian motion, independent of previous related work.

Findings

Identification of a simple phase transition in the model

Use of ODE theory to interpret probabilistic phenomena

Demonstration of the complementarity between probability and ODE analysis

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.