Rigorous derivation of the mean-field limit for the signal-dependent Keller-Segel system

TL;DR

This paper rigorously derives a two-dimensional Keller-Segel system with signal-dependent sensitivity from a stochastic particle model, improving convergence results and establishing an algebraic propagation of chaos rate.

Contribution

It introduces a novel derivation of the Keller-Segel system with enhanced convergence analysis using stopping times and relative-entropy methods.

Findings

Proves convergence of particle system to mean-field limit in probability.

Establishes an algebraic convergence rate for propagation of chaos.

Improves upon existing results with a better scaling regime.

Abstract

We rigorously derive a two-dimensional Keller-Segel type system with signal-dependent sensitivity from a stochastic interacting particle model. By employing suitably defined stopping times, we prove that the convergence of the interacting particle system towards the corresponding mean-field limit equations in probability under an algebraic scaling regime which improves upon existing results with logarithmic scaling. Building on this, we apply the relative-entropy method to obtain strong $L^1$ propagation of chaos, and establish an algebraic convergence rate.