  The existence and the uniqueness of solutions to some semilinear parabolic
equations on homogeneous Lie groups, namely, the Fokker-Planck equation and the
Hamilton-Jacobi equation, are addressed. The anisotropic geometry of the state
space plays a crucial role in our analysis and creates several issues that need
to be overcome. Indeed, the ellipticity directions span, at any point,
subspaces of dimension strictly less than the dimension of the state space.
Finally, the above results are used to obtain the short-time existence of
classical solutions to the mean field games system defined on an homogenous Lie
group.
