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 σ-stable like behaviour. Included are σ-stable processes and fractional Laplace diffusion operators (−Δ)σ2, 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 σ∈(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.
Highlights
Jakobsen their own state and the distribution of the states of the other players, and the mean field game system arises as a characterisation of Nash equilibria when the number of players tends to infinity under certain symmetry assumptions
What differs from the standard Mean Field Games (MFGs) formulation is the type of noise used in the model
Their results have been published in [16]. They consider time-depending MFG systems on the torus with fractional Laplace diffusions and nonlocal couplings. Since they assume additional convexity and coercivity assumptions to ensure global in time semiconcavity and Lipcshitz bounds on solutions, they consider fractional Laplacians of the full range of orders σ ∈ (0, 2)
Summary
In the case of Gaussian noise and local MGF systems, this type of MFG problems with nonlocal or local coupling have been studied from the start [31,32,33,9] and today there is an extensive literature summarized e.g. in [1,23,7] and references therein. Fractional parabolic Bertrand and Carnout MFGs are studied in the recent paper [24] These problems are posed in one space dimension, they have a different and more complicated structure than ours, and the principal terms are the (local) second derivative terms. Their results have been published in [16] They consider time-depending MFG systems on the torus with fractional Laplace diffusions and nonlocal couplings. Since they assume additional convexity and coercivity assumptions to ensure global in time semiconcavity and Lipcshitz bounds on solutions, they consider fractional Laplacians of the full range of orders σ ∈ (0, 2). We give results for local couplings, which in view of the discussion above is a non-trivial extension
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