The global well-posedness for master equations of mean field games of controls

TL;DR

This paper proves the global well-posedness of master equations in mean field games of controls, using monotonicity conditions and Lipschitz continuity to handle the joint law interaction.

Contribution

It establishes the first comprehensive proof of well-posedness for these equations under two key monotonicity conditions, expanding theoretical understanding.

Findings

Proves global well-posedness under Lasry-Lions monotonicity.

Proves global well-posedness under displacement λ-monotonicity.

Analyzes the relation between differential and integral monotonicity conditions.

Abstract

In this manuscript, we establish the global well-posedness for master equations of mean field games of controls, where the interaction is through the joint law of the state and control. Our results are proved under two different conditions: the Lasry-Lions monotonicity and the displacement $\lambda$-monotonicity, both considered in their integral forms. We provide a detailed analysis of both the differential and integral versions of these monotonicity conditions for the corresponding nonseparable Hamiltonian and examine their relation. The proof of global well-posedness relies on the propagation of these monotonicity conditions in their integral forms and a priori uniform Lipschitz continuity of the solution with respect to the measure variable.