Mass transfers and Optimal Stochastic Transports

我的研究计划有三个主要要点。第一个是一个相对较新的方向，专注于随机最优传输问题和具有自由结束时间的动态平均场博弈理论。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，而第三个是我关于涉及哈代-薛定谔算子的边界椭圆边值问题的长期研究项目的延续。 最优鞅大众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，它模拟了一只股票价格的演变。然而，在更高维度的情况下，即“多股票”模型，存在很多悬而未决的问题，其中的问题更具挑战性。布朗鞅的例子特别有趣，因为它将这些主题与布朗运动的最优停止问题和 Skorokhod 嵌入的经典理论联系起来。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新方向，其中优化是针对具有规定结束分布的停止时间和过程。 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，平均场博弈理论独立起源于经济学、工程学和数学分析。该主题可以粗略地描述为对大量小型交互主体中的战略决策的研究。从数学上讲，当玩家数量变得非常大时，它可以被描述为多人纳什均衡的极限。近年来，固定结束时间的平均场地比赛一直是深入研究的主题。然而，仅在几个简单的例子中考虑了具有自由结束时间的平均场博弈，我们的重点将放在这种情况上，其中包含以前的模型尚未捕获的新现象。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。将积极追求对非紧凑环境的重要扩展。多个概率分布（而不是对）之间的多线性传递理论也在开发中。提议者和他的合作者的一个长期研究项目涉及欧几里得空间域上二阶哈代-薛定谔算子的精细分析，此时奇点（零）位于所研究域的边界或其内部。下一阶段将重点关注涉及双曲流形及其分数对应项的 Hardy-Schrodinger 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.