Grushin‐type degenerate parabolic equations with applications to mean field games

In this paper, we establish a weak maximum principle (WMP) and global a priori Schauder estimates for a class of linear degenerate Cauchy problems associated with Grushin‐type operators. These regularity results yield the existence and uniqueness of solutions to the Cauchy problems on the unbounded domain. To derive the WMP, we construct a subsolution tailored to the specific structure of Grushin‐type operators, rather than relying on fundamental solutions. To address the non‐commutativity and inhomogeneity of the Hörmander vector fields, we separate the coefficients via a Taylor expansion at the degenerate points and analyze each component individually. This technique provides an alternative method for the regularity analysis of degenerate operators, which is distinct from the classical framework based on fundamental solutions. As an application, we obtain the existence of classical solutions to a class of degenerate mean field games, in which a typical player may encounter a “forbidden” direction at certain points.