Weak Solution and Invariant Probability Measure for McKean-Vlasov SDEs with Integrable Drifts

TL;DR

This paper establishes the existence and uniqueness of invariant probability measures for McKean-Vlasov SDEs with integrable drifts, using Harnack inequalities, fixed point theorems, and regularity analysis.

Contribution

It introduces new methods to prove weak well-posedness and invariant measure properties for McKean-Vlasov SDEs with integrable drifts, expanding theoretical understanding.

Findings

Proved weak well-posedness using Wang's Harnack inequality and Banach fixed point theorem.

Established regularity properties of invariant measures, including entropy and Sobolev estimates.

Demonstrated existence and uniqueness of invariant measures for symmetric McKean-Vlasov SDEs and stochastic Hamiltonian systems.

Abstract

In this paper, by utilizing Wang's Harnack inequality with power and the Banach fixed point theorem, the weak well-posedness for McKean-Vlasov SDEs with integrable drift is investigated. In addition, using the decoupled method, some regularity such as relative entropy and Sobolev's estimate of invariant probability measure are proved. Finally, by Banach's fixed theorem, the existence and uniqueness of invariant probability measure for symmetric McKean-Vlasov SDEs and stochastic Hamiltonian system with integrable drifts are obtained.