Mass transfers and Optimal Stochastic Transports

我的研究提案有三个主要推动力。第一个是一个相对较新的方向，主要研究随机最优运输问题和具有自由结束时间的动态平均场对策理论。第二个是质量传递的渐近理论，这是申请人最近引入的一个包罗万象的概念；第三个是我长期研究涉及Hardy-Schrodinger算子的椭圆边值问题的项目的继续。最优鞅质量传输理论起源于数学金融，现在人们在一维情况下很好地理解它，在一维情况下，它只对一只股票的价格演变进行建模。然而，在更高维度的情况下，也就是“多股票”模型中，有很多悬而未决的问题，那里的问题更具挑战性。布朗鞅的例子特别有趣，因为它将这些主题与布朗运动的最优停止问题和经典的Skorokhod嵌入理论联系起来。这反过来又导致了具有自由结束时间的最优随机运输的相对较新的方向，其中优化结束于停车时间和具有指定结束分布的过程。随机质量输运理论的欧拉公式与平均场博弈理论密切相关，平均场博弈理论独立于经济学、工程学和数学分析领域。这个主题可以粗略地描述为在非常大的小交互作用的群体中进行战略决策的研究。在数学上，它可以被描述为多人纳什均衡的极限，因为玩家的数量变得非常大。具有固定结束时间的平均场上游戏近年来一直是深入研究的主题。然而，具有自由结束时间的平均场上游戏只被考虑在几个简单的例子中，我们的重点将放在这个例子上，它包含了以前模型没有捕捉到的新现象。另一组相关问题源于质量传递的一般理论及其相关的非线性Kantorovich算子的渐近性质。与大偏差理论的自然联系，以及对符号动力学的应用正在开发中。将积极寻求对非紧凑环境的非平凡扩展。几个概率分布之间的多线性转移理论--而不是成对的--也在发展中。作者和他的合作者的一个长期研究项目是关于欧氏空间中区域上二阶Hardy-Schrodinger算子的精细分析，当奇点(零)位于被研究区域的边界上或其内部时。下一阶段将集中讨论双曲流形上含有Hardy-Schrodinger算子的偏微分方程解的存在性、多解性和定性性质，以及它们的分数阶对应的解的存在性、多解性和定性性质。

There are three main thrusts to my research proposal. The first is a relatively new direction focused on stochastic optimal transport problems and the theory of dynamical mean field games with free end-time. The second is an asymptotic theory of mass transfers, an encompassing concept introduced recently by the applicant, while the third is a continuation of my long-term research project on borderline elliptic boundary value problems involving the Hardy-Schrodinger operator. The theory of optimal martingale mass transport originated in mathematical finance and is now well understood in the one-dimensional case, where it models the evolution of the price of only one stock. However, open questions abound in the higher dimensional case, i.e., for "multi-stock" models, where the problems are much more challenging. The case of Brownian martingales is particularly interesting as it connects these topics to problems of optimal stopping of Brownian motion and the classical theory of Skorokhod embeddings. This in turn leads to the relatively new direction of optimal stochastic transportation with free end time, where the optimization is over stopping times and processes with prescribed end distributions. The Eulerian formulation of stochastic mass transport theory is closely connected to the theory of mean field games, which originated independently in economics, engineering, and mathematical analysis. This topic can be loosely described as the study of strategic decision-making in very large populations of small interacting agents. Mathematically, it can be described as a limit of a multi-player Nash equilibrium as the number of players becomes very large. Mean field games with fixed end times have been the subject of intensive investigations in recent years. However, mean field games with free end time have only been considered in a few simple examples and our focus will be on this case which contains new phenomena that have not been captured by previous models. Another related set of problems stems from the general theory of mass transfers and the asymptotic properties of their associated non-linear Kantorovich operators. Natural connections to large deviation theory, as well as applications to symbolic dynamics are being developed. The non-trivial extensions to a non-compact setting will be actively pursued. A theory of multilinear transfers between several probability distributions -as opposed to pairs- is also under development. A long-term research project of the proposer and his collaborators has been concerned with the fine analysis of second order Hardy-Schrodinger operators on domains in Euclidean space, when the singularity (zero) lies either on the boundary of the domains under study or in their interior. The next phase will focus on issues of existence, multiplicity and qualitative properties of solutions of PDEs involving Hardy-Schrodinger operators on hyperbolic manifolds, as well as their fractional counterparts.