On stationary fractional mean field games

Abstract

We provide an existence result for stationary fractional mean field game systems, with fractional exponent greater than 1/2. In the case in which the coupling is a nonlocal regularizing potential, we obtain existence of solutions under general assumptions on the Hamiltonian. In the case of local coupling, we restrict to the subcritical regime, that is the case in which the diffusion part of the operator dominates the Hamiltonian term. We consider both the case of local bounded coupling and of local unbounded coupling with power-type growth. In this second regime, we impose some conditions on the growth of the coupling and on the growth of the Hamiltonian with respect to the gradient term.

Transport equations with nonlocal diffusion and applications to Hamilton–Jacobi equations

“…Let u be a local weak solution to (2). Assume that ρ is a weak solution to (32). Then, for all ω ∈ (0, τ ) we havê…”

“…We now recall some related results for (1) and (2). As for fractional Fokker-Planck equations, when b ∈ L ∞ or some control on the divergence is assumed, we refer to [32] for stationary problems and to [36] for the evolutive case. Instead, the viscous case is well known, even under weaker assumptions on the velocity field [19,20,37,38,41,62,77,81].…”

“…Related results for Hamilton-Jacobi equations, even degenerate, can be found in [12,15,39,70] and the references therein. Other regularity estimates for space-fractional Fokker-Planck equations, also combined with Hamilton-Jacobi equations in the context of Mean Field Games, can be found in [32,36], while for advection equations with fractional diffusion we refer the reader to [88,89].…”

On an optimal control problem of time-fractional advection-diffusion equation

“…Recently Cirant and Goffi [16] studied a nonlocal (in space) MFG system:      −∂ t u + (−∆) s u + 1 2 |∇u| 2 = f (x, m), (t, x) ∈ (0, T ) × T d ∂ t m + (−∆) s m + div(−m∇u) = 0, m(0, x) = m 0 (x), u(T, x) = u T (x), (31) where (−∆) s denotes the fractional Laplacian of order s, 0 < s < 1. A stationary nonlocal in space MFG system has been studied by Cesaroni et al in [15]. The methods in our paper can be extended to the generalized system: Also, it would be of great interest to construct a mean field game system with both space and time nonlocal structure.…”

On the Existence and Uniqueness of Solutions to Time-Dependent Fractional MFG

“…The stationary counterpart of (1), which heuristically describes an equilibrium state in the long-time regime, has been analyzed very recently by the first author and collaborators [14]. In particular, in [14] the investigation is performed for the subcritical order of the fractional Laplacian s ∈ ( 1 2 , 1), both in the case of local and nonlocal coupling between the equations. There, the well-posedness of the fractional Fokker-Planck equation is based on variational methods, while the study of the fractional HJB equation is established via viscosity solutions' techniques.…”