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The couplings by change of measure are applied to establish log-Harnackinequality(equivalently the entropy-cost estimate) for conditionalMcKean-Vlasov SDEs and derive the quantitative conditional propagation of chaosin relative entropy for mean field interacting particle system with commonnoise. For the log-Harnack inequality, two different types of couplings will beconstructed for non-degenerate conditional McKean-Vlasov SDEs withmultiplicative noise. As to the quantitative conditional propagation of chaosin relative entropy, the initial distribution of interacting particle system isallowed to be singular with that of limit equation. The above results are alsoextended to conditional distribution dependent stochastic Hamiltonian system.

Coupling by Change of Measure for Conditional McKean-Vlasov SDEs and Applications