Non-binary branching process and Non-Markovian exploration process

TL;DR

This paper investigates the exploration process of continuous-time non-binary Galton-Watson trees across different regimes, deriving a limit process and a Ray-Knight representation to understand their asymptotic behavior.

Contribution

It introduces a novel analysis of non-binary branching processes in continuous time, establishing their weak limit and a Ray-Knight representation.

Findings

Derived the weak limit of the exploration process as population size grows

Established a Ray-Knight type representation for the limiting process

Analyzed the process across subcritical, critical, and supercritical regimes

Abstract

We study the exploration (or height) process of a continuous time non-binary Galton-Watson random tree, in the subcritical, critical and supercritical cases. Thus we consider the branching process in continuous time (Z_{t})_{t\geq 0}, which describes the number of offspring alive at time t. We then renormalize our branching process and exploration process, and take the weak limit as the size of the population tends to infinity. Finally we deduce a Ray-Knight representation.