[Paper Review] Binary branching processes with Moran type interactions

Alexander M. G. Cox, E Horton+1 more|arXiv (Cornell University)|Jul 7, 2022

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper introduces a novel binary branching particle system with Moran-type interactions, where particle deaths trigger reproductions and branchings induce deaths, enabling size-constrained population dynamics. The key contribution is an explicit Feynman-Kac-type duality linking the particle system's occupation measure to the underlying Markov process, with an L² convergence rate to the true process as population size increases, outperforming fixed-size Moran models in approximating birth-death processes.

ABSTRACT

The aim of this paper is to study the large population limit of a binary branching particle system with Moran type interactions: we introduce a new model where particles evolve, reproduce and die independently and, with a probability that may depend on the configuration of the whole system, the death of a particle may trigger the reproduction of another particle, while a branching event may trigger the death of an other one. We study the occupation measure of the new model, explicitly relating it to the Feynman-Kac semigroup of the underlying Markov evolution and quantifying the L2 distance between their normalisations. This model extends the fixed size Moran type interacting particle system discussed in [18, 19, 6, 7, 57] and we will indeed show that our model outperforms the latter when used to approximate a birth and death process. We discuss several other applications of our model including the neutron transport equation [36, 15] and population size dynamics.

Motivation & Objective

• To develop a unified particle system model combining binary branching processes with Moran-type resampling to better approximate time-inhomogeneous birth-and-death processes.
• To establish a precise mathematical link between the empirical measure of the particle system and the Feynman-Kac semigroup of the underlying Markov process.
• To quantify the L² distance between the normalized occupation measure of the particle system and the normalized Feynman-Kac semigroup, showing convergence as N → ∞.
• To generalize existing fixed-size Moran models by allowing dynamic population size and configuration-dependent selection/resampling rules.
• To demonstrate the model's superiority over classical fixed-size Moran systems in approximating target processes, particularly in population size dynamics and neutron transport.

Proposed method

• The model introduces a binary branching process where each particle evolves independently via a Markov process X, with branching and killing events occurring at rates b and κ, respectively.
• When a branching or killing event occurs, the system applies configuration-dependent resampling or selection rules: death triggers reproduction, and branching triggers death of another particle.
• The authors define time-dependent weights Π_T^A and Π_T^B to correct for resampling and selection events, explicitly characterizing their impact on the system's empirical measure.
• A key duality is established: the expected value of the weighted empirical measure at time T equals the expectation of f(X_T) weighted by an exponential functional of the cumulative branching and killing rates.
• The L² distance between the normalized occupation measure of the particle system and the normalized Feynman-Kac semigroup is bounded and shown to converge to zero as the initial particle count N → ∞.
• The model is applied to a constrained population size regime (N_min to N_max), where selection/resampling probabilities are set to 1 at boundaries to maintain size control.

Experimental results

Research questions

• RQ1 How can a binary branching particle system be extended with configuration-dependent resampling to maintain population size while preserving convergence to the target process?
• RQ2 What is the precise mathematical relationship between the empirical measure of such a particle system and the Feynman-Kac semigroup of the underlying Markov process?
• RQ3 Can the L² distance between the normalized occupation measure of the particle system and the normalized Feynman-Kac semigroup be explicitly bounded and shown to converge to zero as N → ∞?
• RQ4 Does this model outperform classical fixed-size Moran models in approximating birth-and-death processes, especially in terms of convergence rate and bias?
• RQ5 Under what conditions does the particle system inherit the strong Markov property from the underlying process X?

Key findings

• The expected value of the weighted empirical measure of the particle system at time T is exactly equal to the Feynman-Kac expectation of f(X_T) with a time-dependent exponential weight involving b and κ.
• The L² distance between the normalized occupation measure of the particle system and the normalized Feynman-Kac semigroup converges to zero as the number of initial particles N → ∞.
• The model recovers both the classical many-to-one formula and the unbiased estimator from [57] for fixed-size Moran processes as special cases by choosing appropriate parameters.
• The constrained N_min–N_max model ensures population size remains within bounds by setting selection/resampling probabilities to 1 at boundaries, enabling stable simulation of size-constrained processes.
• The model outperforms classical fixed-size Moran systems in approximating birth-and-death processes, particularly in scenarios with non-constant population sizes or time-inhomogeneous dynamics.
• The particle system's law satisfies a Markov property with respect to the filtration generated by the underlying process, and under regularity conditions, the system inherits the strong Markov property.