Pathwise Regularisation of Singular Interacting Particle Systems and their Mean Field Limits

TL;DR

This paper demonstrates how adding irregular noise paths can regularize singular interacting particle systems, enabling the derivation of their mean field limits under broad conditions on the interaction kernel.

Contribution

It introduces a novel regularization technique using irregular noise paths to handle singular interactions in particle systems and establishes the McKean--Vlasov limit under minimal assumptions.

Findings

Irregular noise can regularize singular particle dynamics.

The McKean--Vlasov limit is recoverable with broad interaction kernel conditions.

The approach accommodates common and idiosyncratic noise sources.

Abstract

We investigate the regularizing effect of certain perturbations by noise in singular interacting particle systems under the mean field scaling. In particular, we show that the addition of a suitably irregular path can regularise these dynamics and we recover the McKean--Vlasov limit under very broad assumptions on the interaction kernel; only requiring it to be controlled in a possibly distributional Besov space. In the particle system we include two sources of randomness, a common noise path $Z$ which regularises the dynamics and a family of idiosyncratic noises, which we only assume to converge in mean field scaling to a representative noise in the McKean--Vlasov equation.