Master Equations for Mean Field Game of Controls: A Unification of Weak Solution Notions
Abstract
In this manuscript, we study the master equation for mean field game of controls, assuming only that the data are Lipschitz continuous in the measure variable. Accordingly, we propose a weaker notion of solution called weak solutions for the master equation. We establish the global well-posedness of the master equation for such solution under the Lasry-Lions monotonicity and displacement λ-monotonicity conditions, respectively. The arguments rely on the analysis of the associated Pontryagin forward-backward stochastic differential equation system, especially the stability in the measure variable. Finally, we review several existing notions of non-smooth solution from the literature, namely good solutions, weak viscosity solutions, Lipschitz solutions, monotone solutions, and examine their relationship with our weak solution. We show that under appropriate assumptions all of them are equivalent by their definitions, and thus our manuscript provides a unification of weak solution notions for the master equations derived from mean field game of controls.
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BibTeX
@misc{li2026master,
title = {{Master Equations for Mean Field Game of Controls: A Unification of Weak Solution Notions}},
author = {Mengzhen Li and Xintian Liu and Chenchen Mou and Zhen Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16524},
eprint = {2609.16524},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16524},
}
RIS
TY - UNPB
TI - Master Equations for Mean Field Game of Controls: A Unification of Weak Solution Notions
AU - Mengzhen Li
AU - Xintian Liu
AU - Chenchen Mou
AU - Zhen Wu
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.16524
UR - https://arxiv.org/abs/2609.16524
AB - In this manuscript, we study the master equation for mean field game of controls, assuming only that the data are Lipschitz continuous in the measure variable. Accordingly, we propose a weaker notion of solution called weak solutions for the master equation. We establish the global well-posedness of the master equation for such solution under the Lasry-Lions monotonicity and displacement λ-monotonicity conditions, respectively. The arguments rely on the analysis of the associated Pontryagin forward-backward stochastic differential equation system, especially the stability in the measure variable. Finally, we review several existing notions of non-smooth solution from the literature, namely good solutions, weak viscosity solutions, Lipschitz solutions, monotone solutions, and examine their relationship with our weak solution. We show that under appropriate assumptions all of them are equivalent by their definitions, and thus our manuscript provides a unification of weak solution notions for the master equations derived from mean field game of controls.
ER -