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

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. Under general assumptions on non-convex confining and interaction potentials with polynomial growth, we establish the well-posedness of the multi-species SDE system, prove propagation of chaos, deriving the corresponding 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. For quadratic interaction potentials and under an appropriate structural assumption, which implies that the generator of each species is multiple of a common generator, we show the existence and (non-) uniqueness of stationary solutions, study their linear stability and prove the existence of a phase transition at low noise strengths. For quadratic and symmetric interaction potentials (but no structural assumption), we construct a free-energy functional that plays the role of a Lyapunov function for the mean-field PDE system. Furthermore, we establish the convergence of solutions to the mean-field PDEs (and of their free energy) to a stationary state (and the corresponding free energy).