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Relative entropy and modulated free energy without confinement via self-similar transformation

Matthew Rosenzweig, Sylvia Serfaty

Abstract

This note extends the modulated entropy and free energy methods for proving mean-field limits/propagation of chaos to the whole space without any confining potential, in contrast to previous work limited to the torus or requiring confinement in the whole space, for all log/Riesz flows. Our novel idea is a scale transformation, sometimes called self-similar coordinates in the PDE literature, which converts the problem to one with a quadratic confining potential, up to a time-dependent renormalization of the interaction potential. In these self-similar coordinates, one can then establish a Grönwall relation for the relative entropy or modulated free energy, conditional on bounds for the Hessian of the mean-field log density. This generalizes recent work of Feng-Wang arXiv:2310.05156, which extended the Jabin-Wang relative entropy method to the whole space for the viscous vortex model. Moreover, in contrast to previous work, our approach allows to obtain uniform-in-time propagation of chaos and even polynomial-in-time generation of chaos in the whole space without confinement, provided one has suitable decay estimates for the mean-field log density. The desired regularity bounds and decay estimates are the subject of a companion paper.

Relative entropy and modulated free energy without confinement via self-similar transformation

Abstract

This note extends the modulated entropy and free energy methods for proving mean-field limits/propagation of chaos to the whole space without any confining potential, in contrast to previous work limited to the torus or requiring confinement in the whole space, for all log/Riesz flows. Our novel idea is a scale transformation, sometimes called self-similar coordinates in the PDE literature, which converts the problem to one with a quadratic confining potential, up to a time-dependent renormalization of the interaction potential. In these self-similar coordinates, one can then establish a Grönwall relation for the relative entropy or modulated free energy, conditional on bounds for the Hessian of the mean-field log density. This generalizes recent work of Feng-Wang arXiv:2310.05156, which extended the Jabin-Wang relative entropy method to the whole space for the viscous vortex model. Moreover, in contrast to previous work, our approach allows to obtain uniform-in-time propagation of chaos and even polynomial-in-time generation of chaos in the whole space without confinement, provided one has suitable decay estimates for the mean-field log density. The desired regularity bounds and decay estimates are the subject of a companion paper.
Paper Structure (14 sections, 12 theorems, 118 equations)

This paper contains 14 sections, 12 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Modulated energy, entropy, and free energy
  3. The self-similar coordinates
  4. Main result
  5. Regularity estimates
  6. Organization of paper
  7. Acknowledgments
  8. Transformation to self-similar coordinates
  9. Mean-field and forward Kolmogorov equations
  10. Relative entropy
  11. Modulated (free) energy
  12. Main proof
  13. Relative entropy
  14. Modulated free energy

Key Result

Theorem 1

Suppose that equation eq:MFlimss admits a solution $\bar{\mu}\in C_w([0,T], \mathcal{P}({\mathbb{R}}^\mathsf{d}))$, the space of functions on $[0,T]$ taking values in $\mathcal{P}({\mathbb{R}}^\mathsf{d})$ and continuous with respect to weak convergence, such that $\|\nabla^{\otimes 2}\log\frac{\bar where $\mathsf{o}_N^t(1)$ is a correction to obtain a nonnegative quantity and depends on $\mu^t$ o

Theorems & Definitions (29)

  • Remark 1.1
  • Theorem 1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6: Added in proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 19 more