Coupled McKean-Vlasov diffusions: wellposedness, propagation of chaos and invariant measures

TL;DR

This paper analyzes a coupled two-species stochastic model influenced by multiple forces, establishing well-posedness, propagation of chaos, and invariant measures, with applications in biological and pedestrian dynamics.

Contribution

It introduces a novel coupled McKean-Vlasov SDE framework and proves key properties including well-posedness, chaos propagation, and invariant measure existence.

Findings

Proved well-posedness of the coupled SDE system

Established propagation of chaos for the particle approximation

Analyzed existence and non-uniqueness of invariant measures

Abstract

In this paper, we study a two-species model in the form of a coupled system of nonlinear stochastic differential equations (SDEs) that arises from a variety of applications such as aggregation of biological cells and pedestrian movements. The evolution of each process is influenced by four different forces, namely an external force, a self-interacting force, a cross-interacting force and a stochastic noise where the two interactions depend on the laws of the two processes. We also consider a many-particle system and a (nonlinear) partial differential equation (PDE) system that associate to the model. We prove the wellposedness of the SDEs, the propagation of chaos of the particle system, and the existence and (non)-uniqueness of invariant measures of the PDE system.