Bismut formula for Lions derivative of distribution-path dependent SDEs

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