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

Abstract

We introduce a class of fully nonlinear mean field games posed in [0, T ] \times \BbbR d. Wejustify that they are related to controlled local or nonlocal diffusions, and more generally in oursetting, to a new control interpretation involving time change rates of stochastic (L\'evy) processes.The main results are the existence and uniqueness of solutions under general assumptions. Theseresults are applied to nondegenerate equations---including both local second-order and nonlocal withfractional Laplacians. Uniqueness holds under the monotonicity of couplings and convexity of theHamiltonian, but neither monotonicity nor convexity need to be strict. We consider a rich classof nonlocal operators and processes and develop tools to work in the whole space without explicitmoment assumptions.