This work is a series of two articles. The main goal is to rigorously derive the degenerate parabolic-elliptic Keller-Segel system in the sub-critical regime from a moderately interacting stochastic particle system. In the first article, we establish the classical solution theory of the degenerate parabolic-elliptic Keller-Segel system and its non-local version. In the second article, which is the current one, we derive a propagation of chaos result, where the classical solution theory obtained in the first article is used to derive required estimates for the particle system. Due to the degeneracy of the non-linear diffusion and the singular aggregation effect in the system, we perform an approximation of the stochastic particle system by using a cut-offed interacting potential. An additional linear diffusion on the particle level is used as a parabolic regularization of the system. We present the propagation of chaos result with two different types of cut-off scaling, namely logarithmic and algebraic scalings. For the logarithmic scaling the convergence of trajectories is obtained in expectation, while for the algebraic scaling the convergence in the sense of probability is derived. The result with algebraic scaling is deduced by studying the dynamics of a carefully constructed stopped process and applying a generalized version of the law of large numbers. Consequently, the propagation of chaos follows directly from these convergence results and the vanishing viscosity argument of the Keller-Segel system.

翻译：本文是系列文章中的第二篇。主要目标是在次临界状态下，从适度相互作用的随机粒子系统中严格推导出退化抛物-椭圆Keller-Segel系统。在第一篇文章中，我们建立了退化抛物-椭圆Keller-Segel系统及其非局部版本的经典解理论。在本文（第二篇）中，我们推导了混沌传播结果，其中利用第一篇文章中获得的经典解理论来推导粒子系统所需的估计。由于系统中非线性扩散的退化性和奇异聚集效应，我们通过使用截断相互作用势对随机粒子系统进行近似。粒子层面的附加线性扩散被用作系统的抛物正则化。我们提出了两种不同截断尺度（即对数尺度和代数尺度）下的混沌传播结果。对于对数尺度，轨迹的收敛性是在期望意义下获得的；而对于代数尺度，则推导出概率意义下的收敛性。代数尺度下的结果是通过研究精心构建的停时过程的动力学并应用广义大数定律得出的。因此，混沌传播直接源于这些收敛结果以及Keller-Segel系统的消失粘性论证。