The ergodicity of nonlinear McKean-Vlasov stochastic differential equations with common noise

Abstract

This paper focuses on the ergodicity of McKean-Vlasov (MV) stochastic differential equations (SDEs) with common noise (wCN), where coefficients depend on both the state and the measure. A major challenge in this setting is that the underlying Markov operator loses the semigroup property, precluding standard ergodic analyses. To circumvent this issue, we lift the system by considering the joint flow of the solution and its conditional distribution. We first construct a semigroup associated with the measure pair of the solution and its conditional distribution. Under polynomial growth conditions, we prove the existence and uniqueness of the invariant measure for the lifted system by a coupling method and obtain an explicit exponential convergence rate. We subsequently derive the strong law of large numbers by a decoupling approach. Building on these results, we establish the uniform-in-time propagation of chaos for the associated mean-field interacting particle system. Furthermore, we establish the convergence of the distribution of a single particle and the empirical measure of the particle system to the marginals of the invariant measure. Finally, illustrative examples are provided to verify the theoretical findings.