Infinite dimensional analysis, viscosity solutions and applications

本课题主要研究无限维空间中的完全非线性一阶和二阶偏微分方程及其应用。这种偏微分方程解的主要例子是与确定性和随机型偏微分方程的最优控制有关的Hamilton-Jacobi-Bellman型(HJB)方程。无限维空间中的偏微分方程组理论从温和解、正则解、弱解和粘性解的角度进行了研究，并在过去的二十年里确立了自己作为无限维分析的现代工具之一的地位。特别是，除了偏微分方程的最优控制外，这类方程的粘性解理论还在研究无限维扩散的大偏差、债券市场理论和数学金融的其他方面以及随机不变性等领域得到了应用。在这个项目中，主要的研究人员集中在两个新兴的无限维偏微分方程领域，这两个领域是完全开放的。第一类是Hilbert空间中的积分偏微分方程解。目的是建立Hilbert空间中完全非线性一阶和二阶积分偏微分方程粘性解理论，并研究其各种应用。对这类方程的兴趣主要来自于它们与无限维跳扩散过程的联系，特别是与由Levy过程驱动的随机PDE的联系。在应用方面，该理论将被用来发展具有小Levy噪声强度的随机偏微分方程解的大偏差的偏微分方程组方法，并研究由脉冲噪声驱动的债券市场理论产生的无限维Black-Scholes和Black-Scholes-Barenblatt积分-偏微分方程组。第二个重点领域是概率度量空间中的偏微分方程组，由于质量传输理论和抽象梯度流和哈密顿流的最新进展，它似乎终于可以发展了。这个新领域非常有趣和重要，该项目将集中于它在大偏差和统计力学中的应用。该项目包含的其他问题包括使用HJB方程获得偏微分方程组最优控制的充要条件和最大值原理的研究。该项目包含一个开创性的研究计划，旨在开发偏微分方程组和无限维分析的新工具。建议的研究领域包括非线性偏微分方程组、泛函分析、概率、随机过程、随机偏微分方程组、数学金融学、最优控制、博弈论、统计力学、质量传输和变分。特别是，它将为无限维跳跃扩散过程的研究提供分析技术，这些过程用于对可能发生随机和暴力事件的各种现象进行随机建模。除了数学之外，该项目还应该对工程、物理、金融和经济等领域的研究产生影响和刺激。该项目的更广泛影响还将包括吸引和培训研究生和博士后学者。

The research of the project is focused on fully nonlinear first- and second-order partial differential equations (PDE) in infinite dimensional spaces and applications thereof. Primary examples of such PDE are equations of Hamilton-Jacobi-Bellman-type (HJB) that are associated with optimal control of deterministic and stochastic PDE. The theory of PDE in infinite dimensional spaces has been studied from the point of view of mild, regular, weak, and viscosity solutions and has established itself over the last two decades as one of the modern tools of infinite dimensional analysis. In particular, in addition to optimal control of PDE, the theory of viscosity solutions of such equations has found applications in areas such as the study of large deviations of infinite dimensional diffusions, the theory of bond markets and other aspects of mathematical finance, and stochastic invariance. In this project the principal investigator focuses on two emerging areas of infinite dimensional PDE that are wide open. The first is integro-PDE in Hilbert spaces. The goal is to develop a viscosity solution theory for fully nonlinear first- and second-order integro-PDE in Hilbert spaces and study its various applications. The interest in such equations comes primarily from their association with infinite dimensional jump-diffusion processes, in particular with stochastic PDE driven by Levy processes. Regarding applications, the theory will be used to develop a PDE approach to large deviations for solutions of stochastic PDE with small Levy noise intensity and to study infinite dimensional Black-Scholes and Black-Scholes-Barenblatt integro-PDE coming from the theory of bond markets driven by impulsive noise. A second area of emphasis is PDE in the space of probability measures, which finally seems open for development owing to recent advances in the theory of mass transport and abstract gradient and Hamiltonian flows. This new area is extremely interesting and important, and the project will concentrate on its applications to large deviations and statistical mechanics. Other problems contained in the project include the use of HJB equations to obtain necessary and sufficient conditions for optimality for optimal control of PDE and investigations into maximum principles.The project contains a pioneering program of research that is aimed at the development of new tools in partial differential equations and infinite dimensional analysis. The proposed research spans areas as diverse as nonlinear partial differential equations, functional analysis, probability, stochastic processes, stochastic partial differential equations, mathematical finance, optimal control, game theory, statistical mechanics, mass transport, and calculus of variations. In particular, it will provide analytical techniques for the study of infinite dimensional jump diffusion processes that are used in stochastic modeling of various phenomena in which random and violent events can occur. Beyond mathematics, the project should have impact and stimulate research in fields such as engineering, physics, finance, and economics. The broader impacts of the project will also include attracting and training graduate students and postdoctoral scholars.