论文标题
在经验度量的非线性功能上，中央限制定理，并应用于相互作用粒子系统的平均场波动
Central limit theorem over non-linear functionals of empirical measures with applications to the mean-field fluctuation of interacting particle systems
论文作者
论文摘要
在这项工作中，为I.I.D的经验度量的非线性函数提出了中央限制定理的广义版本。随机变量，只要功能性满足相关线性功能导数的一些规律性假设。即使对度量分量有非线性依赖性，这种概括也可以应用于蒙特卡洛方法。我们使用此结果来处理初始化在相互作用扩散的经验度量与它们的平均场限制度量（因为粒子的数量转移到无穷大的情况下）之间的波动的收敛性的贡献。使用主方程来处理与时间演化有关的互补贡献，这是一种涉及L衍生物相对于测量成分的抛物线PDE，这是与线性功能衍生物相关的更强的导数概念。
In this work, a generalised version of the central limit theorem is proposed for nonlinear functionals of the empirical measure of i.i.d. random variables, provided that the functional satisfies some regularity assumptions for the associated linear functional derivative. This generalisation can be applied to Monte-Carlo methods, even when there is a nonlinear dependence on the measure component. We use this result to deal with the contribution of the initialisation in the convergence of the fluctuations between the empirical measure of interacting diffusion and their mean-field limiting measure (as the number of particles goes to infinity), when the dependence on measure is nonlinear. A complementary contribution related to the time evolution is treated using the master equation, a parabolic PDE involving L-derivatives with respect to the measure component, which is a stronger notion of derivative that is nonetheless related to the linear functional derivative.
