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

TL;DR

This paper proves that as the number of particles increases, their empirical distribution converges to the Landau equation's solution, especially for very soft and Coulomb potentials, using advanced probabilistic and information-theoretic techniques.

Contribution

It introduces a novel approach combining tightness, uniqueness, and Fisher information dissipation to analyze the propagation of chaos for the Landau equation with Coulomb interactions.

Findings

Convergence of empirical measures to the Landau equation solution.

High integrability estimates derived from Fisher information dissipation.

Affinity of entropy production and Fisher information dissipation in infinite dimensions.

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.