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

TL;DR

This work extends mean field game theory to discounted infinite-time horizons by formulating and analyzing infinite-time forward–backward SDEs that capture social and representative-player dynamics. It proves well-posedness and continuous dependence for these MF-FBSDEs and establishes a flow property that links individual dynamics to the aggregate equilibrium. It shows that the representative player's value function is a viscosity solution to the elliptic master equation, thus connecting the probabilistic FBSDE representation to the PDE master framework in the infinite horizon setting. Overall, the results generalize finite-time mean field game theory and parabolic master equations to a robust infinite-horizon, discounted regime with rigorous FBSDE- and viscosity-based analysis.

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.

Theorems & Definitions (10)

• Theorem 3.2
• Theorem 4.2
• Proposition 4.3
• Theorem 4.4
• Theorem 5.2
• Lemma 5.3
• Theorem 5.4
• Definition 5.5
• Definition 5.6
• Theorem 5.7