We prove existence and uniqueness of classical solutions of the master equation for mean field game (MFG) systems with fractional and nonlocal diffusions. We cover a large class of Lévy diffusions of order greater than 1, including purely nonlocal, local, and even mixed local-nonlocal operators. In the process we prove refined well-posedness results for the MFG systems; the results include the mixed local-nonlocal case. We also show various auxiliary results on viscous Hamilton–Jacobi equations, linear parabolic equations, and linear forward-backward systems that may be of independent interest. This includes a rigorous treatment of certain equations and systems with data and solutions in the duals of Hölder spaces C^₁ on the whole of R^d. We do not assume existence of any moments for the initial distributions of players. In a future work we will use the results of this paper to prove the convergence of N -player games to mean field games as N.
Jakobsen et al. (Wed,) studied this question.