Rigorous Derivation of the Degenerate Parabolic-Elliptic Keller-Segel System from a Moderately Interacting Stochastic Particle System. Part II Propagation of Chaos

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.