发表机构
School of Mathematics, Harbin Institute of Technology; Department of Mathematics, City University of Hong Kong（哈尔滨工业大学数学学院; 香港城市大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对平均场博弈主方程，通过推导非局部抛物型方程并建立Carleman估计，获得了主场及其测度导数沿特征测度流的终端到内部Lipschitz稳定性估计。
AI 中文摘要
本文研究了平均场博弈中主方程的主场及其关于测度变量的泛函导数的Lipschitz稳定性。本工作的一个关键新颖之处在于推导了一个非局部抛物型偏微分方程，该方程控制着沿特征测度流的泛函测度导数。基于该方程，我们为两个足够正则的经典解所对应的测度导数之差建立了一个Carleman估计。我们还为相应的主场之差建立了一个Carleman估计，从而沿特征测度流得到了一个从终端到内部的Lipschitz稳定性估计。结合相关测度流的稳定性，这些估计被用来控制所产生的源项。然后，我们根据终端数据的差异，得到了主场及其泛函测度导数两者的从终端到内部的Lipschitz稳定性估计。该结果为具有测度变量正则性的主场提供了定量控制，并为平均场博弈主方程中主场及其测度导数的稳定性分析发展了一个基于Carleman的框架。
英文摘要
In this paper, we investigate the Lipschitz stability of the master field and its functional derivative with respect to the measure variable for the master equation in mean field games. A key novelty of this work is the derivation of a nonlocal parabolic partial differential equation governing the functional measure derivative along a characteristic measure flow. Based on this equation, we establish a Carleman estimate for the difference of the measure derivatives associated with two sufficiently regular classical solutions. We also establish a Carleman estimate for the difference of the corresponding master fields, yielding a terminal-to-interior Lipschitz stability estimate along a characteristic measure flow. Together with stability of the associated measure flows, these estimates are used to control the resulting source terms. We then obtain a terminal-to-interior Lipschitz stability estimate for both the master field and its functional measure derivative in terms of the discrepancies of the terminal data. The result provides quantitative control of the master field with measure-variable regularity, and develops a Carleman-based framework for the stability analysis of master fields and their measure derivatives in mean field game master equations.
Comments31 pages
