Propagation of Chaos and Stability for the Nonlinear McKean-Vlasov Stochastic Functional Differential Equations with Common Noise

Past dependence is an unavoidable natural phenomenon for dynamic systems. This paper investigates a class of nonlinear McKean–Vlasov stochastic functional differential equations (MV-SFDEs) with common noise. The well-posedness of the nonlinear MV-SFDEs with common noise is demonstrated through the application of the Banach fixed-point theorem. The conditional propagation of chaos with an explicit convergence rate is studied for the MV-SFDEs with common noise and the corresponding functional interacting particle systems. A Razumikhin theorem for the exponential stability is derived via the Itô formula involved with state and measure. Finally, an example is provided to illustrate the result of the stability.