Mass transfers and Optimal Stochastic Transports

我的研究计划主要有三个方面。第一个是一个相对较新的方向集中在随机最优运输问题和理论的动态平均场游戏与自由结束时间。第二个是一个渐进理论的质量转移，一个包容的概念最近介绍的申请人，而第三个是我的长期研究项目的延续边界椭圆边值问题，涉及哈迪-薛定谔算子。 最优鞅质量传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，在一维情况下，它只模拟了一只股票的价格演变。然而，在更高维的情况下，开放的问题比比皆是，即，对于“多库存”模型，问题更具挑战性。布朗鞅的情况下是特别有趣的，因为它连接这些主题的问题，最佳停止布朗运动和经典理论的Skorokhod嵌入。这反过来又导致了相对较新的方向，最佳的随机运输与自由结束时间，其中的优化是在停止时间和过程与规定的结束分布。 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，后者独立起源于经济学、工程学和数学分析。这一主题可以被松散地描述为在非常大的人口的小互动代理的战略决策的研究。在数学上，它可以被描述为多玩家纳什均衡的极限，因为玩家的数量变得非常大。具有固定结束时间的平均场博弈近年来一直是深入研究的主题。然而，平均场游戏与自由结束时间只被认为是在几个简单的例子，我们的重点将是在这种情况下，它包含了新的现象，还没有被以前的模型。 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。大偏差理论的自然联系，以及符号动力学的应用正在开发中。将积极寻求非紧凑设置的非平凡扩展。一个理论的多线性转移之间的几个概率分布-而不是对-也正在发展中。 本文作者及其合作者的一个长期研究项目是对欧氏空间中区域上二阶Hardy-Schrodinger算子的精细分析，当奇点（零）位于所研究区域的边界或内部时.下一阶段将集中在双曲流形上涉及Hardy-Schrodinger算子的偏微分方程解的存在性，多重性和定性性质的问题，以及他们的分数对应。

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.