Killed path-dependent McKean–Vlasov SDEs for a probabilistic representation of non-conservative Mckean PDEs

A McKean–Vlasov stochastic differential equation subject to killing associated to a regularized non-conservative and path-dependent nonlinear parabolic partial differential equation is studied. The existence and pathwise uniqueness of a strong solution and the regularity properties of its sub-probability law are proved. The density of such a law may be seen as a weak solution of the considered PDE. The well-posedness of the associated particle system is also discussed.