Gradient Mean-Field Dynamics with Measure-Valued States: Well-Posedness, Chaos, and Long-Time Stability

Abstract: We study a stochastic mean-field interacting particle system whose state space is $\Y = \Tt^d \times \cP(U)$, where the first component represents a spatial variable and the second one is a probability measure over a compact metric space $U$. The dynamics are driven by locally Lipschitz drift operators: the spatial component evolves according to a Brownian diffusion, while the measure-valued component is perturbed by a projected cylindrical noise acting in the Arens--Eells space. We first establish existence and uniqueness of strong solutions for both the $N$-particle system and the associated nonlinear McKean--Vlasov equation under locally Lipschitz and linear growth assumptions on the drift coefficients. We then prove propagation of chaos: as $N\to\infty$, the empirical measure converges in expectation in Wasserstein--1 distance towards the unique McKean--Vlasov solution. Further, we investigate exponential convergence of the nonlinear McKean--Vlasov dynamics towards a unique invariant measure.