Quantitative particle approximation of nonlinear Fokker-Planck equations with singular kernel

TL;DR

This paper establishes the convergence of particle systems to solutions of nonlinear Fokker-Planck equations with singular kernels, including important examples like Biot-Savart and Keller-Segel, even allowing for finite-time blow-up.

Contribution

It provides the first quantitative approximation results for PDEs with singular interaction kernels using stochastic particle systems, including well-posedness and convergence proofs.

Findings

Quantitative convergence of empirical measures to PDE solutions.

Well-posedness of McKean-Vlasov SDEs with singular kernels.

Convergence holds even for PDEs with finite-time blow-up.

Abstract

In this work, we study the convergence of the empirical measure of moderately interacting particle systems with singular interaction kernels. First, we prove quantitative convergence of the time marginals of the empirical measure of particle positions towards the solution of the limiting nonlinear Fokker-Planck equation. Second, we prove the well-posedness for the McKean-Vlasov SDE involving such singular kernels and the convergence of the empirical measure towards it (propagation of chaos). Our results only require very weak regularity on the interaction kernel, including the Biot-Savart kernel, and attractive kernels such as Riesz and Keller-Segel kernels in arbitrary dimension. For some of these important examples, this is the first time that a quantitative approximation of the PDE is obtained by means of a stochastic particle system. In particular, this convergence still holds (locally in time) for PDEs exhibiting a blow-up in finite time. The proofs are based on a semigroup approach combined with a fine analysis of the regularity of infinite-dimensional stochastic convolution integrals.