Mean Field Games Master Equations with Non-separable Hamiltonians and Displacement Monotonicity

Abstract

In this manuscript, we propose a structural condition on non-separable Hamiltonians, which we term displacement monotonicity condition, to study second order mean field games master equations.A rate of dissipation of a bilinear form is brought to bear a global (in time) well-posedness theory, based on a-priori uniform Lipschitz estimates on the solution in the measure variable. Displacement monotonicity being sometimes in dichotomy with the widely used Lasry-Lions monotonicity condition, the novelties of this work persist even when restricted to separable Hamiltonians.

Monotone solutions for mean field games master equations : continuous state space and common noise

“…Alternatively, the monotone regime proved to be regularizing in the finite state space case [20,3,4]. More recently, several teams have addressed the issue of defining weak solutions of master equations in several context (which are not the monotone regime) : [10,9] propose ways of selecting a weak solution in finite state space, particularly in the potential case ; [22,13,14] introduce notions of weak solutions of the master equation which do not rely on monotonicity assumptions. Up to this point, no general framework has been proposed.…”

Closed-loop convergence for mean field games with common noise

“…There has been great progress in recent years, beginning with the groundbreaking work [16] which heavily exploited the Lasry-Lions monotonicity condition. Well-posedness of the master equation has since been shown in several settings, including major player models [61], finite state space [8,11], lower regularity [59], degenerate idiosyncratic noise [18], and the recent paper [37] using the alternative weak (or displacement) monotonicity concept originating in [5].…”

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].…”