标题： 布朗性粒子的奇异相互作用的相关性估计 显示英文标题

标题： Correlation estimates for Brownian particles with singular interactions

摘要： 我们研究在平均场尺度下具有奇异两体相互作用和非零扩散的粒子系统。 描述平均场行为修正的经典方法是通过关联函数的分析。 对于有界相互作用，关联性的最优估计是众所周知的：对于所有$m$，$m$-粒子关联函数为$G_{N,m}=O(N^{1-m})$。 然而，对于更奇异的相互作用，这些估计仍然难以达到。 在本工作中，我们基于线性化关联函数开发了一个新框架，使我们能够为仅平方可积相互作用核的系统推导出稳健的界限，首次系统地控制了奇异情况下的关联性。 尽管最初不是最优的，但我们可以使用BBGKY层次结构在事后部分改进这些估计：在有界相互作用的情况下，我们的方法以简化的论证恢复了已知的最优估计。 作为关键应用，我们证明了平均场的Bogolyubov修正的有效性，并证明了经验测度的中心极限定理，首次将这些结果扩展到有界相互作用范围之外。 显示英文摘要

摘要： We study particle systems with singular pairwise interactions and non-vanishing diffusion in the mean-field scaling. A classical approach to describing corrections to mean-field behavior is through the analysis of correlation functions. For bounded interactions, the optimal estimates on correlations are well known: the $m$-particle correlation function is $G_{N,m}=O(N^{1-m})$ for all $m$. Such estimates, however, have remained out of reach for more singular interactions. In this work, we develop a new framework based on linearized correlation functions, which allows us to derive robust bounds for systems with merely square-integrable interaction kernels, providing the first systematic control of correlations in the singular setting. Although at first not optimal, our estimates can be partially refined a posteriori using the BBGKY hierarchy: in the case of bounded interactions, our method recovers the known optimal estimates with a simplified argument. As key applications, we establish the validity of the Bogolyubov correction to mean field and prove a central limit theorem for the empirical measure, extending these results beyond the bounded interaction regime for the first time.