Propagation of chaos for topological interactions by a coupling technique

TL;DR

This paper proves that a particle system with topological interactions converges to a kinetic equation as the number of particles grows large, using a coupling method and Wasserstein metric estimates.

Contribution

It introduces a coupling technique to rigorously demonstrate propagation of chaos for topological particle interactions under minimal regularity assumptions.

Findings

Convergence to kinetic equation in large particle limit

Coupling method effectively estimates process distance

Error rate consistent with law of large numbers

Abstract

We consider a system of particles which interact through a jump process. The jump intensities are functions of the proximity rank of the particles, a type of interaction referred to as topological in the literature. Such interactions have been shown relevant for the modelling of bird flocks. We show that, in the large number of particles limit and under minimal smoothness assumptions on the data, the model converges to a kinetic equation which was derived in earlier works both formally and rigorously under more stringent regularity assumptions. The proof relies on the coupling method which assigns to the particle and limiting processes a joint process posed on the cartesian product of the two configuration spaces of the former processes. By appropriate estimates in a suitable Wasserstein metric, we show that the distance between the two processes tends to zero as the number of particles tends to infinity, with an error typical of the law of large numbers.