Mean Field Games and Master equations

有不同的方法来模拟大量相互作用的粒子或媒介。一种这样的方法是基于微观的观点，在这种观点中，人们想要确定每个粒子在任何给定时间的属性（如位置、速度等）。从数学上研究这种方法的一种可能方法是通过一个耦合的普通/随机微分方程系统，将这些属性作为未知数。然而，实现基于这种方法的数值解算器通常是相当昂贵的，而且考虑到大量的（数十亿和数十亿）未知数，很多时候甚至是不可能的。一种主要在统计物理框架中出现的宏观方法，使用了所谓的“平均场视角”。这通常旨在通过粒子密度（即“云”）的时间演变来描述粒子的行为。这样的模型通常会导致对诸如未知数这样的宏观量的偏微分方程。自从他们的引入（大约在2006年，由Caines- Huang- malham<s:1>和Lasry- Lions独立发起）以来，平均场博弈理论在纯数学和应用数学中都找到了多种应用。它研究大群体中的战略决策，其中个体代理通过一定的平均场量（通过其他代理的密度、速度、控制等）相互作用。它为从量子力学到生物多样性生态学的应用提供了强大的工具，并且已经对社会科学、宏观经济学、股票市场、风险管理和财富分配以及生物系统的模型产生了重大影响。该理论起源于统计物理中的平均场理论，然而，在平均场博弈中，智能体希望找到最优策略。这在某种程度上与物理模型形成了对比，在物理模型中，粒子通常受自然法则的支配。因此，在平均场博弈中，一般目标是找到并描述纳什型均衡配置。本文介绍了平均场博弈的主方程。狮子，它代表了理论的核心。这是一个概率测度空间上的无限维非局部Hamilton- Jacobi- Bellman方程，它编码了平均场博弈中的纳什均衡。其中，它是证明和量化随机博弈中智能体数量趋于无穷大时的平均场极限和混沌传播的有力工具，在应用中具有重要意义，在理论上具有重要意义。主方程的可解性问题引发了该领域的一个重要课题和突出的开放性问题。根据这个方程的性质，一般来说，经典解将在有限时间内分解。因此，在满足所谓的Lasry—Lions单调性条件的数据上，无论是在短时间范围内，还是在特殊的结构条件下，都建立了其可解性。此外，文献中的大多数结果都使用了非简并特质噪声的正则化效应。在本提案中，我们将研究退化主方程在缺乏这种正则化效应或结构假设的情况下，由Lasry- Lions单调性条件施加。相反，我们将依赖于所谓的位移单调性条件，它源于最优运输理论中出现的位移凸性概念。这个条件允许我们以统一的方式研究退化模型的一般类别（有时更接近实际应用）。粗略地说，位移单调性也有助于恢复在缺乏非简并噪声时丢失的重要规律性。该建议既包括纯确定性模型，也包括受常见噪声影响的模型。

There are different approaches to model a large collection of interacting particles or agents. One such approach is based on a microscopical viewpoint, where one wants to determine the attributes (such as position, velocity, etc.) of each individual particle at any given time. A possible way to mathematically study such an approach would be by a system of coupled ordinary/stochastic differential equations, having these attributes as unknowns. However, implementing numerical solvers that are based on this approach is in general quite costly, and many times is even impossible, given the huge number (billions and billions) of unknowns. A macroscopic approach, that mainly arose in the framework of statistical physics, uses a so-called 'mean field perspective'. This in general aims to describe the behaviour of the particles via the time evolution of their density (i.e. as a 'cloud'). Such models typically lead to partial differential equations for such macroscopic quantities as unknowns. Since their introduction (initiated around 2006, independently by Caines--Huang--Malhamé, and Lasry--Lions), the theory of mean field games have found multiple applications both in pure and applied mathematics. It studies strategic decision making in large populations where the individual agents interact via certain mean-field quantities (through the density, velocities, controls, etc. of the other agents). It provides powerful tools for applications ranging from quantum mechanics to biodiversity ecology and it has already had a significant impact on models in social sciences, macroeconomics, stock markets, risk management and wealth distribution, and on biological systems. This theory has its roots in the mean field theory from statistical physics, however, in mean field games the agents would like to find optimal strategies. This is somehow in contrast with physical models, where particles are typically governed by the laws of nature. So, in mean field games, the general goal is to find and characterize Nash-type equilibrium configurations. The master equation in mean field games was introduced by P.-L. Lions and it represents the heart of the theory. This is an infinite dimensional nonlocal Hamilton--Jacobi--Bellman equation on the space of probability measures and it encodes the Nash equilibria in mean field games. Among others, it serves as a powerful tool to prove and quantify the mean field limit and propagation of chaos for stochastic games when the number of agents tends to infinity, which is important in applications as well as of great interest theoretically. The question of solvability of the master equation initiated an important programme and outstanding open problems in the field. By the nature of this equation, it is expected that in general classical solutions will break down in finite time. So, its solvability was established either for short time horizon or under special structural conditions on the data that satisfy the so-called Lasry--Lions monotonicity condition. Also, most of the results in the literature use the regularisation effect of a non-degenerate idiosyncratic noise.In this proposal, we will study degenerate master equations in the lack of such a regularisation effect or structural assumptions imposed by the Lasry--Lions monotonicity condition. Instead, we will rely on the so-called displacement monotonicity condition, that stems from the notion of displacement convexity arising in the theory of optimal transport. This condition allows us to investigate a general class of degenerate models (that are sometimes closer to real life applications), by a unified way. Roughly speaking, displacement monotonicity helps also to restore important regularity properties, which were lost in the lack of the non-degenerate noise. This proposal includes both purely deterministic models and models subject to common noise.