Periodic solutions for McKean-Vlasov SDEs under periodic distribution-dependent Lyapunov conditions
Jun Ma
TL;DR
The work addresses the existence of $T$-periodic solutions for McKean–Vlasov SDEs under $T$-periodic distribution-dependent Lyapunov conditions. It develops a periodic Markov process framework on the augmented state space $\mathbb{R}^d\times\mathcal{P}(\mathbb{R}^d)$ and leverages a coupled MVSDE construction, Krylov–Bogoliubov arguments, and the Lévy–Prohorov metric to establish existence, convergence of periodic solutions, and continuous dependence on parameters. Key contributions include introducing a time-periodic Lyapunov structure that depends on both position and distribution, proving the existence of $T$-periodic probabilities for the underlying transition function, and deriving convergence and stability results for both periodic and stationary (time-homogeneous) settings, with several illustrative examples. This framework provides rigorous insights into long-term cyclic dynamics of mean-field systems and has potential applications in mean-field games and large-population stochastic dynamics with distribution feedback.
Abstract
In this paper, we prove the existence of periodic solutions for McKean-Vlasov SDEs under periodic distribution-dependent Lyapunov conditions, which is obtained by periodic Markov processes with state space $\mathbb R^d\times \mathcal P(\mathbb R^d)$. Here $\mathcal P(\mathbb R^d)$ denotes the space of probability measures on $\mathbb R^d$. In addition, we show the convergence to the periodic solution and the continuous dependence on parameters of periodic solutions for McKean-Vlasov SDEs. Finally, we provide several examples to illustrate our theoretical results.