平均场随机偏微分方程：适定性与定量无维混沌传播
Mean-Field Stochastic PDEs: Well-posedness and Quantitative Dimension-Free Propagation of Chaos
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中文总结 AI 辅助
研究涉及伪单调核的平均场随机偏微分方程，通过引入依赖测度的伪单调算子研究其适定性，在p - 一致凸Banach空间建立定量无维混沌传播并获接近最优收敛速率，揭示平均场极限收敛速率与解空间几何的关系，还应用于相关粒子系统。
中文摘要 AI 辅助
本文研究了一类涉及伪单调核的平均场随机偏微分方程。首先，通过引入依赖测度的伪单调算子概念，在变分框架下研究了其在强意义和弱意义下的适定性，推广了经典的Brézis框架。此外，在p - 一致凸Banach空间中为一般无限维弱相互作用系统建立了定量无维混沌传播，得到了在适当意义下接近最优的收敛速率。结果揭示了新见解：平均场极限的收敛速率本质上由基础解空间的几何结构，特别是其凸性模量决定。作为应用，研究了机器学习和流体力学中出现的几个有限维和无限维相互作用粒子系统。
英文摘要
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 Brézis. 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.
