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

We introduce a class of fully nonlinear mean field games posed in $[0,T]\times\mathbb{R}^d$. We justify that they are related to controlled local or nonlocal diffusions, and more generally in our setting, to a new control interpretation involving time change rates of stochastic (L\'evy) processes. The main results are existence and uniqueness of solutions under general assumptions. These results are applied to non-degenerate equations - including both local second order and nonlocal with fractional Laplacians. Uniqueness holds under monotonicity of couplings and convexity of the Hamiltonian, but neither monotonicity nor convexity need to be strict. We consider a rich class of nonlocal operators and processes and develop tools to work in the whole space without explicit moment assumptions.