Mean-Field Stochastic PDEs: Well-posedness and Quantitative Dimension-Free Propagation of Chaos
This work investigates the mean-field stochastic PDEs involving a class of pseudo-monotone kernels. We first study the well-posedness -- in both the strong and weak sense -- within the variational framework by introducing a notion of measure-dependent pseudo-monotone operators, which generalizes the classical framework due to Br\'{e}zis. Furthermore, we establish the quantitative dimension-free propagation of chaos within a $p$-uniformly convex Banach space for general infinite-dimensional weakly interacting systems, obtaining convergence rates that are near-optimal in a suitable sense. Our results reveal a new insight: the convergence rate of the mean-field limit is intrinsically governed by the geometry of the underlying solution space, specifically its modulus of convexity. As applications, we study several finite- and infinite-dimensional interacting particle systems arising in machine learning and fluid mechanics, including stochastic Stein variational gradient descent, mean-field Allen-Cahn equations, and Lagrangian-averaged Burgers equations.