Mean field games with Hormander diffusions
Abstract
In this paper, we study a class of degenerate mean field games (MFG, for short) systems with Hormander diffusion, in which the typical agent can move only along admissible direction. We establish the well-posedness of the MFG systems in intrinsic Holder spaces, which describes the Nash equilibria for a differential game with infinitely many small players. The analysis builds upon degenerate parabolic equations defined on the tours induced by Hormander vector fields without any group structure, for which we develop a global regularity theory for the first time in this general setting. A central difficulty arises from the anisotropic geometry of the state space, which induces non-commutativity and inhomogeneity of the underlying vector fields. Instead of the classical fundamental solution framework, we present an alternative method to derive a priori Schauder estimates for general Hormander degenerate parabolic equations via Campanato spaces.
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BibTeX
@misc{jiang2026mean,
title = {{Mean field games with Hormander diffusions}},
author = {Yiming Jiang and Jingchuang Ren and Yawei Wei and Jie Xue},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23283},
eprint = {2609.23283},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23283},
}
RIS
TY - UNPB
TI - Mean field games with Hormander diffusions
AU - Yiming Jiang
AU - Jingchuang Ren
AU - Yawei Wei
AU - Jie Xue
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23283
UR - https://arxiv.org/abs/2609.23283
AB - In this paper, we study a class of degenerate mean field games (MFG, for short) systems with Hormander diffusion, in which the typical agent can move only along admissible direction. We establish the well-posedness of the MFG systems in intrinsic Holder spaces, which describes the Nash equilibria for a differential game with infinitely many small players. The analysis builds upon degenerate parabolic equations defined on the tours induced by Hormander vector fields without any group structure, for which we develop a global regularity theory for the first time in this general setting. A central difficulty arises from the anisotropic geometry of the state space, which induces non-commutativity and inhomogeneity of the underlying vector fields. Instead of the classical fundamental solution framework, we present an alternative method to derive a priori Schauder estimates for general Hormander degenerate parabolic equations via Campanato spaces.
ER -