Weak Solutions for First Order Mean Field Games with Local Coupling

Abstract

Existence and uniqueness of a weak solution for first order mean field game systems with local coupling are obtained by variational methods. This solution can be used to devise ǫ−Nash equilibria for deterministic differential games with a finite (but large) number of players. For smooth data, the first component of the weak solution of the MFG system is proved to satisfy (in a viscosity sense) a time-space degenerate elliptic differential equation.

New estimates on the regularity of the pressure in density‐constrained mean field games

“…In the case studied in this article, where the density is forced to stay below a given threshold, the naive estimate on the right‐hand side of the Hamilton–Jacobi equation leads to an bound. The estimates obtained previously in do not allow to deduce regularity of the value function (except if ). However, with what we prove in the present paper, one can deduce the following (see Section 2.3 below for the definition of the dual and primal problem).…”

“…The first one has to do with the regularity of the value function . Indeed, as understood in the works by Cardaliaguet and collaborators , a solution of a Hamilton–Jacobi equation exhibits regularity as soon as the right‐hand side is bounded from below and its positive part lies in . In the aforementioned articles, such an assumption on the right‐hand side was obtained by assuming a moderate growth on the penalization of congestion in the primal problem.…”

Global-in-time regularity via duality for congestion-penalized Mean Field Games

“…If one extends the problem min A to a larger class of admissible functions, then the notion of solution that we find should be weakened. We refer the reader to [6] for a survey about these notions, or to [10,12] for the original papers. In particular we underline that the solutions to the MFG system would involve a BV function u, and the HJ equation would become an inequality in the region {m = 0}.…”

“…If the interaction is non-local (of the form g((K * m)(x)) for an interaction kernel K, so that the effective density perceived by the agents is of the form K(x − y)m(y)dy) then it automatically provides more compactness and regularity, results which are not available for local costs. We cite [10] as the first paper providing rigorous definitions and results for the local case. On the other hand, the non-local case is widely discussed in [9].…”

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

“…Since then it has known an intense research activity, and several authors have developed in detail a variety of problems and new directions of research. Among these we mention numerical methods [2,3,30], finite-state problems [14,20,22], obstacle problems [15], extended mean-field games [18,24], probabilistic methods [10,11], long-time behavior [6,8], weak solutions [7,38], applications to economics and environmental policy [26,30,34] to name only a few. For a detailed account of the recent developments and perspectives, we refer the reader to the survey papers [1,5,16,27], the lectures by Lions [35,36] and the recent book [4].…”