Concentration of ground states in stationary mean-field games systems

Abstract

In this paper we provide the existence of classical solutions to stationary mean field game systems in the whole space R N , with coercive potential and aggregating local coupling, under general conditions on the Hamiltonian. The only structural assumption we make is on the growth at infinity of the coupling term in terms of the growth of the Hamiltonian. This result is obtained using a variational approach based on the analysis of the non-convex energy associated to the system. Finally, we show that in the vanishing viscosity limit mass concentrates around the flattest minima of the potential. We also describe the asymptotic shape of the rescaled solutions in the vanishing viscosity limit, in particular proving the existence of ground states, i.e. classical solutions to mean field game systems in the whole space without potential, and with aggregating coupling.

Comparing the best-reply strategy and mean-field games: The stationary case

“…First assume a unique solution (m, u, λ) to (12) exists, then thanks to remark 3.8, we have m = 1 Z e − 2 σ 2 u , for Z satisfying (13c). Then the triple (u, λ, Z) is clearly a solution to (14). Furthermore, suppose another solution (u , λ , Z ) to ( 14) exists.…”

“…They then perturb the solutions to prove existence in the case of a local dependence on the distribution. Other typical methods of proof use continuation methods [21,23,26], Schauder's fixed point theorem [5,17] or variational approaches through energy minimisation problems [14,20]. In our proof we exploit the linear-quadratic nature of the control.…”

On the Variational Formulation of Some Stationary Second-Order Mean Field Games Systems

“…Even if this growth condition is restrictive but crucial for our arguments, the main interest of this variational technique is that it allows proving the existence of weak solutions of (MFG 1 ) for a rather general class of coupling functions f in a straightforward manner. Indeed, as we will show in section 3, f does not need to be monotone (see also [Cir16,CC17,CGPSM16] for some recent results in this direction), and, moreover, we can prove the existence of solutions of variations of system (MFG 1 ) involving couplings which can also depend on the distributional derivatives of m. As a matter of fact, our results are valid for terms in the right-hand side of the first equation in (MFG 1 ), which can be identified with the Gâteaux derivative DF(m) of a functional F : W 1,q (Ω) → R, which is Gâteaux differentiable and weakly lower semicontinuous. As an example of a class of functionals we can deal with, we can take…”

“…Existence and uniqueness results for system (MFG 1 ) have been investigated by several researchers using PDE techniques, starting with the first papers [LL06,LL07] in the framework of weak solutions. The reader is referred to [Cir15,FG16,BF16,Cir16,BOP16,CC17] for other subsequent results on the existence of weak solutions. In addition, under different assumptions on the coupling function f and the growth of H, the existence and uniqueness of smooth solutions have been analyzed in [GPV14,GM15,PV17].…”

On the existence of oscillating solutions in non-monotone Mean-Field Games

“…System (1) then consists of two decoupled MFG systems for (u 1 , m 1 ) and (u 2 , m 2 ); the latter enjoys uniqueness of solutions, while the former is so called "non-monotone" or "focusing". The "non-monotone" setting has been recently considered in [10,12,15,19], and exhibits a much more complicated behaviour than the monotone one (existence of several solutions, concentration, ...). Since the study of the non-monotone case is still at an early stage, we believe that an analysis of the decoupled system can be of interest on its own.…”