Mean-field limits of Riesz-type singular flows

TL;DR

This paper proves mean-field convergence for particle systems with Riesz-type singular interactions, extending previous methods to more general interactions and including stochastic noise, under certain regularity assumptions.

Contribution

It introduces a robust modulated-energy approach that extends functional inequalities to Riesz interactions, accommodating noise and broader singularities.

Findings

Established mean-field convergence for Riesz interactions

Extended functional inequalities to more general singular kernels

Incorporated multiplicative noise into the mean-field analysis

Abstract

We provide a proof of mean-field convergence of first-order dissipative or conservative dynamics of particles with Riesz-type singular interaction (the model interaction is an inverse power $s$ of the distance for any $0<s<d$) when assuming a certain regularity of the solutions to the limiting evolution equations. It relies on a modulated-energy approach, as introduced in previous works where it was restricted to the Coulomb and super-Coulombic cases. The method is also capable of incorporating multiplicative noise of transport type into the dynamics. It relies in extending functional inequalities of arXiv:1803.08345, arXiv:2011.12180, arXiv:2003.11704 to more general interactions, via a new, robust proof that exploits a certain commutator structure.