On fractional and nonlocal parabolic mean field games in the whole space

[摘要] 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 Levy processes with some sigma-stable like behaviour. Included are sigma-stable processes and fractional Laplace diffusion operators (-Delta)(sigma/2) tempered nonsymmetric processes in Finance, spectrally one-sided processes, and sums of subelliptic op-erators of different orders. Our main results are existence and uniqueness of classical solutions of MFG systems with nondegenerate diffusion operators of order sigma is an element of (1, 2). We consider parabolic equations in the whole space with both local and nonlocal couplings. Our proofs use 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 require no moment assumptions and use a weaker topology than Wasserstein. (C) 2021 The Author(s). Published by Elsevier Inc.

[发布日期] 2021-11-15

[关键词] Mean field games;Fractional PDE;Existence;Uniqueness;Classical solutions;Nonlocal and local couplings