On a strong form of propagation of chaos for McKean-Vlasov equations

TL;DR

This paper strengthens the convergence mode of particle systems to McKean-Vlasov limits, showing empirical measures and fixed particles converge in stronger topologies under minimal assumptions, with concise probabilistic proofs.

Contribution

It introduces a stronger form of propagation of chaos for McKean-Vlasov equations with nondegenerate volatility, using minimal continuity assumptions and probabilistic methods.

Findings

Empirical measure converges in a stronger topology than weak convergence.

Fixed particles converge in total variation to their limit law.

Provides new existence and uniqueness results for McKean-Vlasov equations.

Abstract

This note shows how to considerably strengthen the usual mode of convergence of an $n$-particle system to its McKean-Vlasov limit, often known as propagation of chaos, when the volatility coefficient is nondegenerate and involves no interaction term. Notably, the empirical measure converges in a much stronger topology than weak convergence, and any fixed $k$ particles converge in total variation to their limit law as $n\rightarrow\infty$. This requires minimal continuity for the drift in both the space and measure variables. The proofs are purely probabilistic and rather short, relying on Girsanov's and Sanov's theorems. Along the way, some modest new existence and uniqueness results for McKean-Vlasov equations are derived.