Short time solution to the master equation of a first order mean field game

Abstract

The goal of this paper is to show existence of short-time classical solutions to the so called Master Equation of first order Mean Field Games, which can be thought of as the limit of the corresponding master equation of a stochastic mean field game as the individual noises approach zero. Despite being the equation of an idealistic model, its study is justified as a way of understanding mean field games in which the individual players' randomness is negligible; in this sense it can be compared to the study of ideal fluids. We restrict ourselves to mean field games with smooth coefficients but do not impose any monotonicity conditions on the running and initial costs, and we do not require convexity of the Hamiltonian, thus extending the result of Gangbo and Swiech to a considerably broader class of Hamiltonians.MSC: 34A12, 35R06, 35R15, 45K05, 49L99, 49N70, 91A13, 91A23.

Viability analysis of the first-order mean field games

“…Using inclusion supp(χ) ⊂ SOL(s 2 , s 0 , m(·)) we conclude that χ is concentrated on the set of optimal motions to problem (33), (34).…”

“…x, then it solves master equation (31) in the viscosity sense [13,Proposition 4.20]. However, the existence of the classical solution is proved only for the special case of nondegenerate stochastic mean field game [9] or on the short time interval [13], [19], [34]. Furthermore, since there are examples of multiplicity of solutions to the mean field game [5], the regularity conditions required in [13,Proposition 4.20] are not fulfilled in the general case.…”

Weak solutions of the master equation for Mean Field Games with no idiosyncratic noise

“…Lions also developed the Hilbertian approach [32] in order to handle equation of the form (0.1) or (0.2), which yields the existence of classical solutions under a structure condition on H and F ensuring the convexity of the solution with respect to the space variable. A partial list of references on the master equation is [2,3,5,6,7,8,9,11,17,27,28,33,34].…”

Well-posedness of mean field games master equations involving non-separable local Hamiltonians

“…Depending on the techniques used in these works to show the well-posedness of the corresponding master equations, one may group these results into three possible categories. We refer to a non-exhaustive list of works as follows: probabilistic ideas for problems including individual or common noise were used in [13,12,20,33]; variational techniques (based on optimal transport or optimal control theory in Hilbert spaces, for problems without noise or with individual noise) were exploited in [23,31,19,8]; and finally PDE techniques were used in [11,10] to attack problems with common or individual noise. In most of these references, a special hypothesis is assumed on the Hamiltonian H appearing in the master equation, namely it is such that the momentum variable it is separated from the measure variable, i.e.…”

“…The recent work [20] constructs global in time classical solutions to master equations involving non-separable Hamiltonians, under an additional, so-called displacement monotonicity assumption on the Hamiltonian. Finally, [31] constructs local in time classical solutions to the master equation in the deterministic setting for smooth regularizing Hamiltonians that have the separable structure.…”