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

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