Rank-dependent Galton-Watson processes and their pathwise duals

TL;DR

This paper introduces a modified Galton-Watson process with rank-dependent offspring distributions and explores its pathwise duals using a graphical representation that reveals coalescing trajectories and forest structures.

Contribution

It develops a new framework for rank-dependent Galton-Watson processes and establishes a novel graphical duality representation with coalescing Markov chain trajectories.

Findings

Established a rank-dependent Galton-Watson process model.

Developed a graphical representation of pathwise duality.

Demonstrated coalescence of trajectories forming forest graphs.

Abstract

We introduce a modified Galton-Watson process using the framework of an infinite system of particles labeled by $(x,t)$, where $x$ is the rank of the particle born at time $t$. The key assumption concerning the offspring numbers of different particles is that they are independent, but their distributions may depend on the particle label $(x,t)$. For the associated system of coupled monotone Markov chains, we address the issue of pathwise duality elucidated by a remarkable graphical representation, with the trajectories of the primary Markov chains and their duals coalescing together to form forest graphs on a two-dimensional grid.