A law of large numbers for branching Markov processes by the ergodicity of ancestral lineages

TL;DR

This paper establishes a law of large numbers for structured branching populations by analyzing the ergodic behavior of ancestral lineages, demonstrating convergence of empirical trait distributions to a mean trajectory.

Contribution

It introduces a novel ergodicity-based approach to prove a law of large numbers for ancestral trajectories in branching Markov processes.

Findings

Empirical distributions converge to the mean trait trajectory.

The method applies to size-structured populations with environmental variability.

The approach is demonstrated on a size-structured exponential growth model.

Abstract

We are interested in the dynamic of a structured branching population where the trait of each individual moves according to a Markov process. The rate of division of each individual is a function of its trait and when a branching event occurs, the trait of a descendant at birth depends on the trait of the mother. We prove a law of large numbers for the empirical distribution of ancestral trajectories. It ensures that the empirical measure converges to the mean value of the spine which is a time-inhomogeneous Markov process describing the trait of a typical individual along its ancestral lineage. Our approach relies on ergodicity arguments for this time-inhomogeneous Markov process. We apply this technique on the example of a size-structured population with exponential growth in varying environment.