Derivative Estimates on Distributions of McKean-Vlasov SDEs

TL;DR

This paper develops derivative estimates for the distributions of McKean-Vlasov SDEs using heat kernel expansion, enabling bounds on total variation distance via Wasserstein distance, extending recent distribution-free noise results.

Contribution

It introduces a novel approach using heat kernel expansion to estimate derivatives of the law of McKean-Vlasov SDEs with respect to initial distributions.

Findings

Derived intrinsic derivative estimates for the law of McKean-Vlasov SDEs.

Bounded total variation distance between solution laws by Wasserstein distance.

Extended recent results for distribution-free noise using coupling and Malliavin calculus.

Abstract

By using the heat kernel parameter expansion with respect to the frozen SDEs, the intrinsic derivative is estimated for the law of Mckean-Vlasov SDEs with respect to the initial distribution. As an application, the total variation distance between the laws of two solutions is bounded by the Wasserstein distance for initial distributions. These extend some recent results proved for distribution-free noise by using the coupling method and Malliavin calculus.