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We consider the path-dependent McKean-Vlasov equation, in which both thedrift and the diffusion coefficients are allowed to depend on the wholetrajectory of the process up to the current time t, and depend on thecorresponding marginal distributions. We prove the strong well-posedness of theequation in the L^p setting, p greater or equal than 2, locally in time, aswell as the propagation of chaos properties. Then, we introduce an interpolatedEuler scheme, a key object to simulate numerically the process, and we provethe convergence, with an explicit rate, of this scheme towards the strongsolution in the L^p norm. As applications we give results for two mean-fieldequations arising in biology and neuroscience.

Path-dependent McKean-Vlasov equation: strong well-posedness, propagation of chaos and convergence of an interpolated Euler scheme