Mass transfers and Optimal Stochastic Transports

我的研究计划主要有三个重点。第一个是相对较新的方向，主要研究随机最优输运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，这是申请人最近介绍的一个包含性概念，而第三个是我长期研究项目的延续，涉及Hardy-Schrodinger算子的边线椭圆边值问题。最优鞅质量运输理论起源于数学金融学，现在在一维情况下得到了很好的理解，它只模拟了一只股票的价格演变。然而，在更高维度的情况下，例如，对于“多库存”模型，开放的问题大量存在，其中的问题更具挑战性。布朗鞅的例子特别有趣，因为它将这些话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终点时间的最优随机运输的相对新的方向，其中优化是超过停止时间和具有规定终点分布的过程。随机质量传递理论的欧拉公式与平均场博弈理论密切相关，后者分别起源于经济学、工程学和数学分析。这个主题可以粗略地描述为在非常大的相互作用的小代理群体中的战略决策的研究。从数学上讲，它可以被描述为多玩家纳什均衡的极限，因为玩家数量变得非常大。具有固定结束时间的平均场比赛是近年来深入研究的主题。然而，具有自由结束时间的平均场比赛只在几个简单的例子中被考虑过，我们的重点将放在这种情况上，它包含了以前模型没有捕捉到的新现象。另一组相关的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，以及符号动力学的应用正在发展中。将积极追求非紧凑设置的非平凡扩展。在几个概率分布（而不是成对分布）之间的多线性转移理论也在发展中。在欧几里得空间中，当奇异点（零）位于所研究域的边界或内部时，二阶Hardy-Schrodinger算子的精细分析是作者及其合作者的一个长期研究项目。下一阶段将重点研究双曲流形上Hardy-Schrodinger算子及其分数阶对应算子的偏微分方程解的存在性、多重性和定性性质问题。

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.