Mean-field limit of a particle approximation for the parabolic-parabolic Keller-Segel model

TL;DR

This paper rigorously proves the convergence of a stochastic particle system to the mean-field parabolic-parabolic Keller-Segel equation, demonstrating propagation of chaos in any dimension with a specific cut-off parameter.

Contribution

It establishes the propagation of chaos for the Keller-Segel particle system with a logarithmic cut-off, providing a rigorous convergence analysis to the mean-field PDE.

Findings

Joint distribution becomes $f$-chaotic as N→∞

The measure $f$ is a weak solution to the Keller-Segel PDE

Convergence holds for any spatial dimension

Abstract

In this paper, we study propagation of chaos for the parabolic-parabolic Keller-Segel model with a logarithmic cut-off by establishing a rigorous convergence analysis from a stochastic particle system to the parabolic-parabolic Keller-Segel (KS) equation for any dimension case. Under the assumption that the initial data are independent and identically distributed (i.i.d.) with a common probability density function $\rho_0$, we rigorously prove the propagation of chaos for this interacting system with a cut-off parameter $\varepsilon\sim (\ln N)^{-\frac{2}{d+2}}$: when $N\rightarrow \infty$, the joint distribution of the particle system is $f$-chaotic and the measure $f$ possesses a density which is a weak solution to the mean-field parabolic-parabolic KS equation.