A free boundary problem for the mean-field limit of diffusing particles with nonlinear boundary reactivity

TL;DR

This paper establishes a rigorous mean-field limit for a system of diffusing particles with nonlinear boundary reactivity, leading to a free boundary problem that generalizes classical boundary conditions.

Contribution

It introduces a novel mean-field framework for particles with reactive boundaries, extending classical models to include nonlinear and nonlocal reactivity effects.

Findings

Proves existence and uniqueness of the mean-field limit

Identifies the limit as a free boundary problem with nonlinear reactivity

Provides a rigorous probabilistic analysis of epidemic spreading models

Abstract

Consider a finite system of diffusing particles coupled through a reactive boundary. Each particle is reflected, but may react with the boundary according to a killing mechanism which depends on the current reactivity of the boundary and the particle's local time along it. With every such reaction, the boundary moves and its reactivity adjusts. We show that this system admits a unique mean-field limit, described by a free boundary problem with nonlinear and nonlocal reactivity. The latter generalises the classical Robin condition for the case of a fixed boundary with constant reactivity. Via Skorokhod's M1 topology and a characterisation of the particles' behaviour near the boundary, we first identify the weak limit points of the empirical measure flows with killing. Then, we combine a probabilistic decoupling technique and energy estimates to prove uniqueness and deduce convergence. Our analysis gives a rigorous mean-field description of a model of epidemic spreading. Moreover, it contributes to the literature on inert drift systems and yields a novel mean-field perspective on the recent encounter-based framework for diffusion-mediated surface reaction from [Phys. Rev. Lett. 125 (2020) 078102].