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Propagation of chaos for the Landau equation with very soft and Coulomb potentials

Côme Tabary

TL;DR

This work proves propagation of chaos for the space-homogeneous Landau equation with very soft and Coulomb potentials ($\gamma\in[-3,-2]$) by analyzing a regularized $N$-particle system and passing to the mean-field limit. The authors develop a robust tightness/uniqueness strategy based on entropy and (generalized) Fisher information dissipation, introducing new infinite-dimensional affinity results that extend classical functionals to level-3 quantities. Key technical pillars include uniform bounds and dissipation estimates for the particle system, an infinite-dimensional mean-field framework, and a conditional $L^1_tL^\infty_v$ control that enables uniqueness of the limit via existing Landau-weak-solution results. The combination yields convergence of the empirical measure to the classical Landau solution $g_t$ and convergence of all $j$-particle marginals to $g_t^{\otimes j}$, thereby establishing propagation of chaos in the very soft/Coulomb regime. The methods and infinite-dimensional affinity results may have broader implications for mean-field limits in kinetic equations with singular interactions.

Abstract

We consider a drift-diffusion process of $N$ stochastic particles and show that its empirical measure converges, as $N\rightarrow \infty$, to the solution of the Landau equation. We work in the regime of very soft and Coulomb potentials using a tightness/uniqueness method. To claim uniqueness, we need high integrability estimates that we obtain by crucially exploiting the dissipation of the Fisher information at the level of the particle system. To be able to exploit these estimates as $N\rightarrow \infty$, we prove the affinity in infinite dimension of the entropy production and Fisher information dissipation (and other first and second-order versions of the Fisher information through a general theorem), results which were up to now only known for the entropy and the usual Fisher information.

Propagation of chaos for the Landau equation with very soft and Coulomb potentials

TL;DR

This work proves propagation of chaos for the space-homogeneous Landau equation with very soft and Coulomb potentials () by analyzing a regularized -particle system and passing to the mean-field limit. The authors develop a robust tightness/uniqueness strategy based on entropy and (generalized) Fisher information dissipation, introducing new infinite-dimensional affinity results that extend classical functionals to level-3 quantities. Key technical pillars include uniform bounds and dissipation estimates for the particle system, an infinite-dimensional mean-field framework, and a conditional control that enables uniqueness of the limit via existing Landau-weak-solution results. The combination yields convergence of the empirical measure to the classical Landau solution and convergence of all -particle marginals to , thereby establishing propagation of chaos in the very soft/Coulomb regime. The methods and infinite-dimensional affinity results may have broader implications for mean-field limits in kinetic equations with singular interactions.

Abstract

We consider a drift-diffusion process of stochastic particles and show that its empirical measure converges, as , to the solution of the Landau equation. We work in the regime of very soft and Coulomb potentials using a tightness/uniqueness method. To claim uniqueness, we need high integrability estimates that we obtain by crucially exploiting the dissipation of the Fisher information at the level of the particle system. To be able to exploit these estimates as , we prove the affinity in infinite dimension of the entropy production and Fisher information dissipation (and other first and second-order versions of the Fisher information through a general theorem), results which were up to now only known for the entropy and the usual Fisher information.

Paper Structure

This paper contains 24 sections, 44 theorems, 343 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Propagation of chaos
  5. Generalized Fisher information
  6. Literature review and discussion
  7. Notations and layout of the proof
  8. Preliminaries on Fisher information
  9. The regularized particle system
  10. Entropy and Fisher information in the Kolmogorov equation
  11. Tightness of the particle system
  12. Infinite-dimension limit of the entropy and Fisher information estimates
  13. The cluster points only charge almost sure weak solutions
  14. A final $L^1_t L^\infty_v$ estimate and uniqueness of the cluster point
  15. A conditional $L^1_t L^\infty_v$ bound
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $\gamma\in[-3,-2]$. Let $g_0\in L^1(\mathbb{R}^3)$ have unit mass, finite energy $E_0$, finite entropy $H_0$, finite Fisher information $I_0$, and finite moment of order $m$ for some $m> 3(2-\gamma)$ ; and let $(g_t)_{t\geq 0}$ be the classical solution to the Landau equation eq:landau with initial condition $g_0$. Then, for any $T>0$, the random empirical measure converges in probability in $

Theorems & Definitions (120)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Proposition 2.1
  • ...and 110 more