Stability of the numerical scheme for stochastic McKean-Vlasov equations

This paper studies the infinite-time stability of the numerical scheme for stochastic McKean-Vlasov equations (SMVEs) via stochastic particle method. The long-time propagation of chaos in mean-square sense is obtained, with which the almost sure propagation in infinite horizon is proved by exploiting the Chebyshev inequality and the Borel-Cantelli lemma. Then the mean-square and almost sure exponential stabilities of the Euler-Maruyama scheme associated with the corresponding interacting particle system are shown through an ingenious manipulation of empirical measure. Combining the assertions enables the numerical solutions to reproduce the stabilities of the original SMVEs. The examples are demonstrated to reveal the importance of this study.