Mean Field Limits of Particle-Based Stochastic Reaction-Diffusion Models

TL;DR

This paper derives deterministic mean field PDE models from particle-based stochastic reaction-diffusion models, providing a rigorous convergence proof in the large-population limit for biological systems.

Contribution

It introduces a measure-valued stochastic process framework and proves convergence to PDEs, bridging stochastic particle models and deterministic equations in reaction-diffusion systems.

Findings

Derived coarse-grained PDE models from stochastic particle models.

Established consistency with Doi Fock Space representation.

Proved convergence in the large-population limit.

Abstract

Particle-based stochastic reaction-diffusion (PBSRD) models are a popular approach for studying biological systems involving both noise in the reaction process and diffusive transport. In this work we derive coarse-grained deterministic partial integro-differential equation (PIDE) models that provide a mean field approximation to the volume reactivity PBSRD model, a model commonly used for studying cellular processes. We formulate a weak measure-valued stochastic process (MVSP) representation for the volume reactivity PBSRD model, demonstrating for a simplified but representative system that it is consistent with the commonly used Doi Fock Space representation of the corresponding forward equation. We then prove the convergence of the general volume reactivity model MVSP to the mean field PIDEs in the large-population (i.e. thermodynamic) limit.