Periodic solutions for McKean-Vlasov SDEs under periodic distribution-dependent Lyapunov conditions

TL;DR

This paper establishes the existence, convergence, and parameter dependence of periodic solutions for McKean-Vlasov SDEs using periodic Lyapunov conditions and Markov process techniques.

Contribution

It introduces a novel approach to analyze periodic solutions of McKean-Vlasov SDEs under distribution-dependent Lyapunov conditions, including existence and stability results.

Findings

Existence of periodic solutions under Lyapunov conditions

Convergence to periodic solutions over time

Continuous dependence on parameters

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.