Periodic solutions to Mckean–Vlasov SDEs under Lyapunov conditions

Abstract In this article, we investigate the existence of periodic solutions to McKean–Vlasov stochastic differential equations subject to periodic Lyapunov conditions with distributional dependence. The proof is based on the construction of periodic Markov processes on the product space R n × P ( R n ) ${\mathbb{R}}^{n}{\times}\mathcal{P}\left({\mathbb{R}}^{n}\right)$ , where P ( R n ) $\mathcal{P}\left({\mathbb{R}}^{n}\right)$ is the space of probability measures on R n ${\mathbb{R}}^{n}$ . Moreover, we also prove the existence of periodic solutions under Lyapunov conditions, where the Lyapunov functions involve only the spatial components. To illustrate our analysis, we present several concrete examples.