Particle approximation of the doubly parabolic Keller-Segel equation in the plane

TL;DR

This paper develops a particle approximation method for the doubly parabolic Keller-Segel equation in the plane, demonstrating existence, tightness, and convergence of the particle system to the PDE solution under certain conditions.

Contribution

It introduces a novel particle system for the parabolic-parabolic Keller-Segel model and proves its convergence to the PDE solution via a Markovianization technique.

Findings

Existence of the particle system for small sensitivity parameter.

Tightness of the empirical measure as particle number increases.

Weak convergence of the empirical measure to the Keller-Segel PDE solution.

Abstract

In this work, we study a stochastic system of N particles associated with the parabolic-parabolic Keller-Segel system. This particle system is singular and non Markovian in that its drift term depends on the past of the particles. When the sensitivity parameter is sufficiently small, we show that this particle system indeed exists for any $N\geq 2$, we show tightness in $N$ of its empirical measure, and that any weak limit point of this empirical measure, as $N\to\infty$, solves some nonlinear martingale problem, which in particular implies that its family of time-marginals solves the parabolic-parabolic Keller-Segel system in some weak sense. The main argument of the proof consists of a Markovianization of the interaction kernel: We show that, in some loose sense, the two-by-two path-dependant interaction can be controlled by a two-by-two Coulomb interaction, as in the parabolic-elliptic case.