To characterize the regularity of distribution-path dependent SDEs in initial distributions variable as probability measures on the path space, we introduce the intrinsic and Lions derivatives in the space of probability measures on Banach spaces, and prove the chain rule for the Lions derivative in the distribution of Banach-valued random variables. By using Malliavin calculus, we establish the Bismut type formula for the Lions derivatives of functional solutions to SDEs with distribution-path dependent drifts. When the noise term is also path dependent so that the Bismut formula is invalid, we establish the asymptotic Bismut formula. Both nondegenerate and degenerate noises are considered. The main results of this talk generalize and improve the corresponding ones derived recently in the literature for the classical SDEs with memory and McKean-Vlasov SDEs without memory.

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