Multi-species McKean-Vlasov dynamics in non-convex landscapes

TL;DR

This paper analyzes multi-species stochastic particle systems and their mean-field PDEs in complex landscapes, focusing on long-term behavior, stability, phase transitions, and the existence of stationary solutions.

Contribution

It introduces a comprehensive analysis of multi-species McKean-Vlasov systems in non-convex landscapes, including well-posedness, stability, and phase transition phenomena.

Findings

Existence and (non-)uniqueness of stationary solutions

Construction of a free-energy functional as a Lyapunov function

Identification of phase transitions at low noise levels

Abstract

In this paper, we study multi-species stochastic interacting particle systems and their mean-field McKean-Vlasov partial differential equations (PDEs) in non-convex landscapes. We discuss the well-posedness of the multi-species SDE system, propagation of chaos and the derivation of the coupled McKean-Vlasov PDE system in the mean field limit. Our focus is on the long-time and asymptotic behaviour of the mean-field PDEs. Under suitable growth assumptions on the potentials and an appropriate structural assumption, we show the existence and (non-) uniqueness of stationary solutions and study their linear stability. Under a symmetry assumption we construct a free-energy functional that plays the role of a Lyapunov function for the mean-field PDE system. Furthermore, we prove the existence of a phase transition at low noise strengths and establish the convergence of solutions to the mean-field PDEs (and of their free energy) to a stationary state (and the corresponding free energy).