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

Li Chen, Veniamin Gvozdik, Yue Li

TL;DR

This work rigorously derives the subcritical degenerate parabolic-elliptic Keller-Segel system from moderately interacting stochastic particles on $\mathbb{R}^d$ ($d\ge3$). It uses mollified interactions and parabolic regularization to tame degeneracy and aggregation, pairing a generalized intermediate particle model with a generalized particle model and associated PDEs, then proves propagation of chaos under two scalings: a logarithmic cut-off (in expectation) and an algebraic cut-off (in probability). The authors establish both convergence in expectation with explicit rates and convergence in probability with sharp rates, leveraging a stopped-process framework, duality, and a relative-entropy approach to obtain strong propagation of chaos results in $L^1$ and $L^q$ norms. The results connect the microscopic particle dynamics to the macroscopic KS dynamics via vanishing mollification and viscosity, with quantitative estimates that underpin the validity of mean-field limits in degenerate, non-local aggregation-diffusion systems.

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.

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 work rigorously derives the subcritical degenerate parabolic-elliptic Keller-Segel system from moderately interacting stochastic particles on (). It uses mollified interactions and parabolic regularization to tame degeneracy and aggregation, pairing a generalized intermediate particle model with a generalized particle model and associated PDEs, then proves propagation of chaos under two scalings: a logarithmic cut-off (in expectation) and an algebraic cut-off (in probability). The authors establish both convergence in expectation with explicit rates and convergence in probability with sharp rates, leveraging a stopped-process framework, duality, and a relative-entropy approach to obtain strong propagation of chaos results in and norms. The results connect the microscopic particle dynamics to the macroscopic KS dynamics via vanishing mollification and viscosity, with quantitative estimates that underpin the validity of mean-field limits in degenerate, non-local aggregation-diffusion systems.

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.
Paper Structure (5 sections, 13 theorems, 183 equations)

This paper contains 5 sections, 13 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Particle models \ref{['generalized_intermediate_particle_model']} and \ref{['generalized_particle_model']}
  3. Convergence in Expectation with the Logarithmic Cut-off
  4. Convergence in Probability with the Algebraic cut-off
  5. Propagation of chaos in the strong sense

Key Result

Theorem 1

Assume that $T>0$, $m=2$ or $m\geq 3$, $\sigma\geq 0$, and Assumption ass holds, then the problem generalized_particle_model has a unique strong solution $\hat{X}_t^{i, \sigma}$ with $u^\sigma(t,\cdot)$ as the density of its law, and furthermore it holds

Theorems & Definitions (28)

  • Theorem 1
  • Remark 2
  • Corollary 3: Propagation of chaos in the weak sense
  • Remark 4
  • Theorem 5: Quantitative propagation of chaos result in $L^q(\mathbb{R}^{dl})$-norm
  • Proposition 6: Solvability of problem \ref{['generalized_particle_model']}
  • proof
  • Proposition 7: Solvability of the problem \ref{['generalized_intermediate_particle_model']}
  • Proposition 8
  • proof
  • ...and 18 more