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.

翻译：本文通过随机粒子方法研究随机McKean-Vlasov方程（SMVEs）数值格式的无限时稳定性。首先，获得了均方意义下的长期混沌传播性质，并利用Chebyshev不等式与Borel-Cantelli引理证明了无限时域上的几乎必然传播。随后，通过对经验测度的巧妙处理，展示了与相应相互作用粒子系统关联的Euler-Maruyama格式的均方指数稳定性与几乎必然指数稳定性。结合上述结论，数值解能够重现原始SMVEs的稳定性。最后，通过实例展现了本研究的重要性。