Rate of homogenization for fully-coupled McKean-Vlasov SDEs

TL;DR

This paper investigates the homogenization rates of fully-coupled McKean-Vlasov stochastic differential equations with ergodic fast components, providing convergence rates without periodicity assumptions.

Contribution

It introduces new convergence rate results for fully-coupled McKean-Vlasov SDEs under ergodic conditions, expanding homogenization theory beyond periodic settings.

Findings

Derived explicit convergence rates to homogenized limits.

Established ergodic theorems for McKean-Vlasov systems.

Analyzed regularity of Poisson equations on Wasserstein space.

Abstract

We consider a fully-coupled slow-fast system of McKean-Vlasov SDEs with full dependence on the slow and fast component and on the law of the slow component and derive convergence rates to its homogenized limit. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. In the course of the proof we obtain related ergodic theorems and we gain results on the regularity of Poisson type of equations and of the associated Cauchy-Problem on the Wasserstein space that are of independent interest.