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Quantitative relative entropy estimates on the whole space for convolution interaction forces

Paul Nikolaev, David J. Prömel

Abstract

Quantitative estimates are derived, on the whole space, for the relative entropy between the joint law of random interacting particles and the tensorized law at the limiting systeme. The developed method combines the relative entropy method under the moderated interaction scaling introduced by Oeschläger, and the propagation of chaos in probability. The result includes the case that the interaction force does not need to be a potential field. Furthermore, if the interaction force is a potential field, with a convolutional structure, then the developed estimate also provides the modulated energy estimates. Moreover, we demonstrate propagation of chaos for large stochastic systems of interacting particles and discuss possible applications to various interacting particle systems, including the Coulomb interaction case.

Quantitative relative entropy estimates on the whole space for convolution interaction forces

Abstract

Quantitative estimates are derived, on the whole space, for the relative entropy between the joint law of random interacting particles and the tensorized law at the limiting systeme. The developed method combines the relative entropy method under the moderated interaction scaling introduced by Oeschläger, and the propagation of chaos in probability. The result includes the case that the interaction force does not need to be a potential field. Furthermore, if the interaction force is a potential field, with a convolutional structure, then the developed estimate also provides the modulated energy estimates. Moreover, we demonstrate propagation of chaos for large stochastic systems of interacting particles and discuss possible applications to various interacting particle systems, including the Coulomb interaction case.
Paper Structure (15 sections, 18 theorems, 144 equations)

This paper contains 15 sections, 18 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Problem setting, preliminaries and main results
  3. Particle systems
  4. Associated PDEs
  5. Preliminary results
  6. Main results:
  7. Relative entropy method
  8. Relative entropy and modulated energy
  9. L2-estimate
  10. Special Choices of W and V
  11. De-regularization of the high dimensional PDE and the limiting PDE
  12. Application
  13. Uniform bounded confidence model
  14. Parabolic-Elliptic Keller--Segel System
  15. Parabolic-Elliptic Keller--Segel System with Bessel potential

Key Result

Theorem 2.9

Let $\rho^{N,\varepsilon}$ and $\rho^{\varepsilon}$ be the non-negative solutions of eq: regularized_Liouville_equation and of eq: regularized_aggregation_diffusion_pde respectively. Assume that the convergence in probability, Assumption ass: convergence_in_probability, and the law of large numbers, where $\rho^{N,2,\varepsilon}$ is the $2$-marginal density of $\rho^{N,\varepsilon}$. Furthermore,

Theorems & Definitions (46)

  • Definition 2.1: Weak solutions
  • Definition 2.2: Weak solutions
  • Remark 2.3
  • Definition 2.7
  • Remark 2.8
  • Theorem 2.9
  • Remark 2.10
  • Theorem 2.11
  • Remark 2.12
  • Lemma 3.1
  • ...and 36 more