Estimates for MFGs

In the absence of special transformations or explicit solutions, the analysis of MFGs often relies on a priori bounds. Here, we investigate estimates that are commonly used. We begin by using the maximum principle to obtain one-sided bounds. Next, we consider energy-type estimates that give additional bounds. These two techniques extend to a broad class of mean-field game problems. Equally important are the consequences of these bounds when combined with earlier results. In Sect. 6.3, we develop some of these aspects. In the remainder of this chapter, we discuss other methods that rely on the particular structure of the problems. First, we present a second-order estimate that is used frequently in the periodic setting. Next, we consider a technique that gives Lipschitz bounds for stationary first-order MFGs. Subsequently, we examine energy conservation principles. Finally, we prove estimates for the Fokker–Planck equation that depend on uniform ellipticity or parabolicity of the MFG system.