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

Abstract

In this paper we construct short time classical solutions to a class of master equations in the presence of non-degenerate individual noise arising in the theory of mean field games. The considered Hamiltonians are non-separable and local functions of the measure variable, therefore the equation is restricted to absolutely continuous measures whose densities lie in suitable Sobolev spaces. Our results hold for smooth enough Hamiltonians, without any additional structural conditions as convexity or monotonicity.

Monotone Solutions of the Master Equation for Mean Field Games with 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].…”

Policy iteration method for time-dependent Mean Field Games systems with non-separable Hamiltonians

“…[4]). From the PDE point of view, the short time existence of solutions to general MFGs with non-separable Hamiltonians has been studied in [29,25], a series of papers by Ambrose et al [12,13,14] and Gangbo et al in [32]. The probabilistic approach to this problem has been considered in [28].…”

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].…”