Calculus of Variations and Hamilton-Jacobi Equations in the Wasserstein Space

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本计画的目的是研究Wasserstein机率测度空间上的一类动力系统，其对应于流体力学与量子力学中的一些基本偏微分方程系统。Wasserstein空间方法的一个很好的特性是它能够处理奇异数据和奇异解。此外，也有可能处理离散和连续的模型与相同的形式主义。该计划的核心在于发展一个适当的理论变分法和哈密尔顿-雅可比方程在瓦瑟斯坦空间。在这个项目中，主要研究人员将研究以下主题：最小化在概率测度的Wasserstein空间中的绝对连续路径集合上定义的某些物理上有意义的拉格朗日函数的作用泛函; Wasserstein空间中Hamilton-Jacobi方程粘性解的存在性和唯一性;以及这些Hamilton-Jacobi方程的均匀化。这些问题中最具挑战性的方面是概率测度空间中缺乏局部紧性，以及拉格朗日算子和哈密尔顿算子中存在强奇异性。变分法和哈密尔顿-雅可比方程在物理学、经济学和工程学等许多科学领域中发挥着核心作用。特别是，它们是经典力学的主要研究对象。预计本项目所研究的变分问题和Hamilton-Jacobi方程的解的性质将为理解流体力学和量子力学中某些无限维动力系统的动力学提供重要的一步。首席研究员的瓦瑟斯坦形式主义方法也有望给出一个统一的框架，在这个框架中，粒子的经典和量子行为都可以被描述。拟议的活动将提供一个富有成效的互动与一些其他学科的理论和应用数学，如最佳质量运输，双曲系统的守恒律，概率论和无限维最优控制理论。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This project aims to study a class of dynamical systems on the Wasserstein space of probability measures corresponding to some fundamental systems of partial differential equations in fluid and quantum mechanics. A nice feature of the Wasserstein space approach is its ability to handle singular data and singular solutions. In addition, there is also a possibility of handling discrete and continuous models with the same formalism. The core of this program lies in the development of an appropriate theory for calculus of variations and Hamilton-Jacobi equations in the Wasserstein space. In this project, the principal investigator will investigate the following topics: minimizing the action functional for certain physically meaningful Lagrangians defined on the set of absolutely continuous paths in the Wasserstein space of probability measures; existence and uniqueness of viscosity solutions for Hamilton-Jacobi equations in the Wasserstein space; and homogenization of these Hamilton-Jacobi equations. Some of the most challenging aspects to these problems are the lack of local compactness in the space of probability measures and the presence of strong singularities in the Lagrangians and Hamiltonians.Calculus of variations and Hamilton-Jacobi equations play a central role in many areas of sciences including physics, economics, and engineering. In particular, they are the main object of study in classical mechanics. It is expected that the properties of solutions of the variational problems and the Hamilton-Jacobi equations to be studied in this project will provide a significant step in understanding the dynamics of some infinite-dimensional dynamical systems arising in fluid and quantum mechanics. The principal investigator's Wasserstein formalism approach is also expected to give a unifying framework in which both classical and quantum behavior of particles can be described. The proposed activity will provide a fruitful interaction with some other disciplines of theoretical and applied mathematics such as optimal mass transportation, hyperbolic systems of conservation laws, probability theory, and infinite-dimensional optimal control theory.