Infinite-time Mean Field FBSDEs and Viscosity Solutions to Elliptic Master Equations

Abstract: This paper presents a further investigation of the properties of infinite-time mean field FBSDEs and elliptic master equations, which were introduced in \cite{yang2025discounted} as mathematical tools for solving discounted infinite-time mean field games. By establishing the continuous dependence of the FBSDE solutions on their initial values, we prove the flow property of the mean field FBSDEs. Furthermore, we prove that, at the Nash equilibrium, the value function of the representative player constitutes a viscosity solution to the corresponding elliptic master equation. Our work extends the classical theory of finite-time mean field games and parabolic master equations to the infinite-time setting.