On fractional and nonlocal parabolic Mean Field Games in the whole space

Abstract

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 Lévy 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.

Major update. A much larger class of anomalous diffusions, local coupling results are now complete, problems are posed in the whole space and not the compact torus. New equicontinuity, $L^1$, and $L^{\infty }$-results for the Fokker-planck equation are needed and proved by using pure PDE arguments. We now work in a more general metrics than Wasserstein