Strong convergence of propagation of chaos for McKean-Vlasov SDEs with singular interactions

TL;DR

This paper proves the strong convergence of particle systems approximating McKean-Vlasov SDEs with singular interactions, extending well-posedness results and achieving optimal convergence rates using advanced analytical techniques.

Contribution

It extends strong well-posedness results to mixed $L^p$-drifts and establishes optimal convergence rates for particle approximations with singular interaction kernels.

Findings

Proved strong convergence of particle systems with singular interactions.

Extended well-posedness results to mixed $L^p$-drifts.

Achieved optimal convergence rates using entropy methods and Zvonkin's transformation.

Abstract

In this work we show the strong convergence of propagation of chaos for the particle approximation of McKean-Vlasov SDEs with singular $L^p$-interactions as well as for the moderate interaction particle systems on the level of particle trajectories. One of the main obstacles is to establish the strong well-posedness of the SDEs for particle systems with singular interaction. To this end, we extend the results on strong well-posedness of Krylov and R\"ockner \cite{Kr-Ro} to the case of mixed $L^p$-drifts, where the heat kernel estimates play a crucial role. Moreover, when the interaction kernel is bounded measurable, we also obtain the optimal rate of strong convergence, which is partially based on Jabin and Wang's entropy method \cite{JW16} and Zvonkin's transformation.