In this talk, we are concerned with McKean-Vlasov SDEs with memory, where the drift coefficients are monotone w.r.t. the spatial variables. By the Malliavin calculus, we establish the Bismut formulas for Lions derivatives of Markov operators associated with functional solutions when the diffusions are path-independent. Also, we establish the asymptotic Bismut formulas for Lions derivatives of Markov operators corresponding to functional solutions not only for the non-degenerate stochastic dynamical systems but also for degenerate counterparts (e.g., stochastic Hamiltonian systems) provided that the diffusion terms are path-dependent.
Moreover, we introduce the intrinsic and Lions derivatives for probability measures on a separable Banach space, and derive the chain rule for functionals of the distributions of Banach-valued random variables, which is also interesting in their right.

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