Poisson equation on Wasserstein space and diffusion approximations for McKean-Vlasov equation

TL;DR

This paper investigates the asymptotic behavior of multi-time-scale McKean-Vlasov equations by analyzing a Poisson equation on Wasserstein space, revealing new homogenized drift terms and providing convergence estimates.

Contribution

It introduces a novel analysis of the Poisson equation on Wasserstein space for McKean-Vlasov systems with multiple time scales, including new homogenized drift terms.

Findings

Derived asymptotic limits for slow processes.

Established quantitative error estimates for convergence.

Identified new homogenized drift terms involving measure derivatives.

Abstract

We consider the fully-coupled McKean-Vlasov equation with multi-time-scale potentials, and all the coefficients depend on the distributions of both the slow component and the fast motion. By studying the smoothness of the solution of the non-linear Poisson equation on Wasserstein space, we derive the asymptotic limit as well as the quantitative error estimate of the convergence for the slow process. Extra homogenized drift term containing derivative in the measure argument of the solution of the Poisson equation appears in the limit, which seems to be new and is unique for systems involving the fast distribution.