On fully nonlinear parabolic mean field games with examples of nonlocal and local diffusions.

We prove existence and uniqueness of solutions of a class of abstract fully nonlinear mean field game systems. We justify that such problems are related to controlled local or nonlocal diffusions, or more specifically, controlled time change rates of stochastic (L\'evy) processes. Both the system of equations and the precise control interpretation seem to be new. We show that the results apply for degenerate equations of low fractional order, and some nondegenerate equations, local and nonlocal.