[Paper Review] A law of large numbers for branching Markov processes by the ergodicity of ancestral lineages

Aline Marguet|arXiv (Cornell University)|Jul 22, 2017

Stochastic processes and statistical mechanics25 references21 citations

TL;DR

This paper establishes a law of large numbers for branching Markov processes by leveraging the ergodicity of ancestral lineages, showing that the empirical distribution of ancestral trajectories converges to the mean of a time-inhomogeneous Markov process describing the typical lineage. The approach avoids spectral methods and applies to time-inhomogeneous dynamics, including a size-structured population with exponential growth in a varying environment.

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.

Motivation & Objective

• To establish a law of large numbers for the empirical measure of ancestral trajectories in a structured branching Markov process.
• To analyze the asymptotic behavior of a population where individual traits evolve via a Markov process and influence division rates.
• To develop a method that avoids spectral theory and instead relies on ergodicity of the spine process for convergence results.
• To extend convergence results to time-inhomogeneous dynamics, particularly in varying environmental conditions.
• To provide a foundation for statistical inference in population models, such as estimating division rates in cell dynamics.

Proposed method

• Uses spinal techniques to characterize the trait evolution along a typical ancestral lineage as a time-inhomogeneous Markov process.
• Applies ergodicity arguments to the spine process to establish convergence of the empirical measure.
• Employs the Many-to-One formula to relate the first moment semigroup to the behavior of the empirical measure.
• Derives bounds on the first harmonic moment of the auxiliary process using Kolmogorov's forward equation and Grönwall's inequality.
• Establishes uniform integrability and moment bounds via Itô’s formula and exponential moment estimates.
• Verifies key assumptions (A–F) using explicit moment bounds and comparison arguments for the branching rate and transition kernels.

Experimental results

Research questions

• RQ1 Does the empirical distribution of ancestral trajectories in a branching Markov process converge to a deterministic limit under time-inhomogeneous dynamics?
• RQ2 Can the convergence be established without relying on spectral theory or eigenelements?
• RQ3 How does the ergodicity of the ancestral lineage process ensure the law of large numbers for the empirical measure?
• RQ4 What conditions guarantee uniform integrability and moment bounds in the time-inhomogeneous case?
• RQ5 Can this method be applied to models with environmental variability, such as size-structured populations with time-dependent growth?

Key findings

• The empirical measure of ancestral trajectories converges almost surely to the mean of the spine process, a time-inhomogeneous Markov process describing the typical lineage.
• The convergence holds under classical assumptions on the Markov process and division rate, without requiring spectral decomposition.
• The first harmonic moment of the auxiliary process is uniformly bounded in time, ensuring stability of the spine measure.
• Moment bounds for the population size and its second moment are established via Itô’s formula and exponential moment estimates.
• The method applies to time-inhomogeneous models, including a size-structured population with exponential growth in a varying environment.
• Uniform integrability of the normalized population size is verified, confirming the validity of the law of large numbers.