  We study Mean Field Games (MFGs) driven by a large class of nonlocal,
fractional and anomalous diffusions in the whole space. These non-Gaussian
diffusions are pure jump L\'evy processes with some $\sigma$-stable like
behaviour. Included are $\sigma$-stable processes and fractional Laplace
diffusion operators $(-\Delta)^{\frac{\sigma}2}$, tempered nonsymmetric
processes in Finance, spectrally one-sided processes, and sums of subelliptic
operators of different orders. Our main results are existence and uniqueness of
classical solutions of MFG systems with nondegenerate diffusion operators of
order $\sigma\in(1,2)$. We consider parabolic equations in the whole space with
both local and nonlocal couplings. Our proofs uses pure PDE-methods and build
on ideas of Lions et al. The new ingredients are fractional heat kernel
estimates, regularity results for fractional Bellman, Fokker-Planck and coupled
Mean Field Game equations, and a priori bounds and compactness of (very) weak
solutions of fractional Fokker-Planck equations in the whole space. Our
techniques requires no moment assumptions and uses a weaker topology than
Wasserstein.
