A Class of Degenerate Mean Field Games, Associated FBSDEs, and Master Equations

In this paper, we study a class of degenerate mean field games (MFGs) with state-distribution dependent and unbounded functional diffusion coefficients. With a probabilistic method, we study the well-posedness of the forward-backward stochastic differential equations associated with the MFG and arising from the maximum principle, and estimate the corresponding Jacobian and Hessian flows. We further establish the regularity of the value functional [Formula: see text]; in particular, we show that when the cost function is [Formula: see text] in the spatial and control variables and [Formula: see text] in the distribution argument, then the value functional is [Formula: see text] in time and [Formula: see text] in the spatial and distribution variables. As a consequence, the value functional [Formula: see text] is the unique classical solution of the degenerate MFG master equation.