Infinite-Time Mean Field FBSDEs and the Associated Elliptic Master Equations

This paper presents a further investigation of the properties of infinite-time mean field forward-backward stochastic differential equations (FBSDEs) and the associated elliptic master equations, which were introduced in [18] 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. And then, we prove that, at the Nash equilibrium, the value function of the representative player constitutes a viscosity solution to the corresponding elliptic master equation. In particular, when the coefficients of the equations are distribution-independent, we construct a classical solution to the elliptic partial differential equation (PDE) via fully coupled infinite-time FBSDEs. Furthermore, for classical solutions possessing displacement monotonicity and certain growth conditions, we establish their uniqueness for the elliptic master equation.