Mean Field Stochastic Partial Differential Equations with Nonlinear Kernels

Abstract: This work focuses on the mean field stochastic partial differential equations with nonlinear kernels. We first prove the existence and uniqueness of strong and weak solutions for mean field stochastic partial differential equations in the variational framework, then establish the convergence (in certain Wasserstein metric) of the empirical laws of interacting systems to the law of solutions of mean field equations, as the number of particles tends to infinity. The main challenge lies in addressing the inherent interplay between the high nonlinearity of operators and the non-local effect of coefficients that depend on the measure. In particular, we do not need to assume any exponential moment control condition of solutions, which extends the range of the applicability of our results. As applications, we first study a class of finite-dimensional interacting particle systems with polynomial kernels, which are commonly encountered in fields such as the data science and the machine learning. Subsequently, we present several illustrative examples of infinite-dimensional interacting systems with nonlinear kernels, such as the stochastic climate models, the stochastic Allen-Cahn equations, and the stochastic Burgers type equations.