具有Hormander扩散的平均场博弈

中文摘要 AI 辅助

本文研究了一类具有Hormander扩散的退化平均场博弈（MFG，简称）系统，其中典型智能体只能沿允许方向移动。我们在内在Hölder空间中建立了MFG系统的适定性，该系统描述了具有无限多个小参与者的微分博弈的纳什均衡。分析基于由Hormander向量场在环面上诱导的退化抛物方程，且无需任何群结构，我们首次在这种一般背景下发展了全局正则性理论。一个核心困难源于状态空间的各向异性几何，这导致底层向量场的非交换性和非齐次性。我们提出了一种替代经典基本解框架的方法，通过Campanato空间推导一般Hormander退化抛物方程的先验Schauder估计。

英文摘要

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.

发表机构

• School of Mathematical Sciences and LPMC Nankai University（南开大学数学科学学院和LPMC）
• Yau Mathematical Sciences Center Tsinghua University（清华大学丘成桐数学科学中心）
• School of Mathematical Sciences Ocean University of China（中国海洋大学数学科学学院）

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