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

TL;DR

This paper rigorously derives the Keller-Segel system from a stochastic particle model, establishing propagation of chaos with regularization techniques to handle degeneracy and singular aggregation effects.

Contribution

It provides a novel propagation of chaos result for the Keller-Segel system using approximation and regularization methods.

Findings

Propagation of chaos established with logarithmic scalings

Convergence in expectation demonstrated

Regularization techniques effectively handle degeneracy and singularity

Abstract

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 [7], 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 logarithmic scalings. Consequently, the propagation of chaos follows directly from convergence in the sense of expectation and the vanishing viscosity argument of the Keller-Segel system.