Propagation of chaos for maxima of particle systems with mean-field drift interaction

TL;DR

This paper proves that the normalized maxima of large particle systems with mean-field interactions behave like independent particles following McKean--Vlasov dynamics, despite the maxima's dependence on all particles.

Contribution

It establishes propagation of chaos for the maxima of particle systems, a result not directly implied by classical propagation of chaos, using a novel change of measure and combinatorial analysis.

Findings

Normalized maxima follow McKean--Vlasov dynamics in large populations

Propagation of chaos holds for maxima, not just fixed particles

New proof technique involving change of measure and stochastic integrals

Abstract

We study the asymptotic behavior of the normalized maxima of real-valued diffusive particles with mean-field drift interaction. Our main result establishes propagation of chaos: in the large population limit, the normalized maxima behave as those arising in an i.i.d. system where each particle follows the associated McKean--Vlasov limiting dynamics. Because the maximum depends on all particles, our result does not follow from classical propagation of chaos, where convergence to an i.i.d. limit holds for any fixed number of particles but not all particles simultaneously. The proof uses a change of measure argument that depends on a delicate combinatorial analysis of the iterated stochastic integrals appearing in the chaos expansion of the Radon--Nikodym density.