Convergence towards the population cross-diffusion system from stochastic many-particle system

TL;DR

This paper rigorously derives a non-local cross-diffusion system as the limit of a stochastic many-particle system, using mollified interactions and probabilistic convergence techniques.

Contribution

It introduces a novel approach to prove convergence from stochastic particle systems to cross-diffusion PDEs with mollified interactions.

Findings

Established probabilistic convergence in the whole space

Developed an intermediate mollified particle system

Used Taylor expansion and Gronwall estimates for proof

Abstract

In this paper, we derive rigorously a non-local cross-diffusion system from an interacting stochastic many-particle system in the whole space. The convergence is proved in the sense of probability by introducing an intermediate particle system with a mollified interaction potential, where the mollification is of algebraic scaling. The main idea of the proof is to study the time evolution of a stopped process and obtain a Gronwall type estimate by using Taylor's expansion around the limiting stochastic process.