Time-Dependent Mean-Field Games in the Subquadratic Case

Abstract

In this paper we consider time-dependent mean-field games with subquadratic Hamiltonians and power-like local dependence on the measure. We establish existence of classical solutions under a certain set of conditions depending on both the growth of the Hamiltonian and the dimension. This is done by combining regularity estimates for the Hamilton-Jacobi equation based on the Gagliardo-Nirenberg interpolation inequality with polynomial estimates for the Fokker-Planck equation. This technique improves substantially the previous results on the regularity of timedependent mean-field games.D.

Existence of a solution to an equation arising from the theory of Mean Field Games

“…In the past five years alone, a large number of manuscripts have been devoted to it, revealing its importance, impact, and possible applications (see e.g. [7,8,15,[23][24][25][26][27][28]30,31,[33][34][35]39,[41][42][43][49][50][51][52]). In light of the publications [44][45][46][47], we restrict our study to the simplest framework of games: those with identical players.…”

Regularity for Mean-Field Games Systems with Initial-Initial Boundary Conditions: The Subquadratic Case

“…We recover a series of previously obtained results. See [25]. The next lemma is a simple consequence of the Gagliardo-Niremberg interpolation inequality:…”

“…Then, an additional argument based on the non-linear adjoint method, see [13], yields Lipschitz regularity for u . See [25]. Several additional estimates for a solution .u ; m / can be derived once Lipschitz regularity for solutions to the HamiltonJacobi equation is established, as outlined next.…”

“…The first results concerning smooth solutions for Hamiltonians with quadratic and subquadratic growth appeared in [36]. These were substantially improved in [25]. The superquadratic case was treated in [23].…”

Crowd-Averse Robust Mean-Field Games: Approximation via State Space Extension

“…In this sense, a variety of solution schemes has been recently proposed based on discretization and/or numerical approximations [1]. There has been substantial progress in the theory of weak solutions [9] and classical solutions [11], [12] for mean-field games. Robust mean-field games have been formulated in [5], [6].…”