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The well-posedness and exponential ergodicity are proved for stochasticHamiltonian systems containing a singular drift term which is locallyintegrable in the component with noise. As an application, the well-posednessand uniform exponential ergodicity are derived for a class of singulardegenerated McKean-Vlasov SDEs.

Singular Degenerate SDEs: Well-Posedness and Exponential Ergodicity