Levy Processes, Martingales and Spectral Theory

Levy 过程、鞅和谱理论

• 批准号: 1005844
• 负责人: Rodrigo Banuelos
• 金额: $ 33万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-15 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1005844&HistoricalAwards=false
• 关键词: Levy; Processes; Martingales; Spectral; Theory

This project deals with several problems and conjectures which lie at the interface of probability, harmonic analysis and the spectral theory for the Laplacian, the fractional Laplacian, and its relativistic versions. It aims to find new sharp inequalities for a class of Fourier multipliers that result from functional modifications of Levy processes via the Levy-Khintchine formula. It proposes to give universal bounds for the trace of the semigroup of Brownian motion, stable and relativistic stable processes and to explore the second order asymptotics for the spectral counting function for these processes. The unifying theme in these investigations is the use of the symbols of the underlying Levy process. Martingale theory, heat kernel estimates and techniques (perhaps even yet to bediscovered) from the theory of the wave equations for fractional Laplacians, will play a key role in these investigations.Martingales, which were invented more than a hundred years ago to explain the theory of "fair" games, play a fundamental role in many branches of mathematics and their applications in countless areas of the physical, biological and social sciences. They are intimately related to a second order differential operator knows as the Laplacian. This operator, named after the mathematical astronomer Marquis Pierre-Simon De Laplace more than two hundred years ago, gives the matheamtical foundation for the theory of heat, waves, electricity, magnetism and fluids. Both martingales and the Laplace operator interact with Brownian motion which models physical, biological and economic systems, and even with more general Levy processes which are widely used to model various financial systems which are subject to more instantaneous changes or jumps. At the root of the interactions between martingales, the Laplace operator and Brownian motion, is a deep mathematical theory which makes these applications scientifically sound. This project deals with several of these core questions and relates them to several other branches of mathematics including more general partial differential equations and their applications to the diffusion of heat, propagation of waves and their connections with random phenomenon.These projects will involve graduate students. The results will be disseminated through publications in professional journals, lectures and on the web. A sincere effort will be made to expose (and involve) students and young Ph.D.'s from underrepresented groups to this research and to increase their participation in mathematics.

该项目涉及的几个问题和示意图，位于概率，调和分析和拉普拉斯，分数拉普拉斯，及其相对论版本的谱理论的接口。通过Levy-Khintchine公式对Levy过程进行泛函修正，得到一类Fourier乘子的新的尖锐不等式.给出了布朗运动、稳定过程和相对论稳定过程的半群迹的泛界，并讨论了这些过程的谱计数函数的二阶渐近性. 在这些调查的统一主题是使用的符号的基础利维过程。 鞅论，热核估计与技巧（也许甚至还没有被发现）从分数拉普拉斯波动方程理论，将在这些调查中发挥关键作用。鞅，这是发明了一百多年前解释理论的“公平”游戏，发挥了基本作用，在许多分支的数学和他们的应用在无数领域的物理，生物和社会科学。它们与一个被称为拉普拉斯算子的二阶微分算子密切相关。 这个算子是以两百多年前数学天文学家皮埃尔-西蒙·德·拉普拉斯的名字命名的，它为热、波、电、磁和流体的理论提供了数学基础。鞅和拉普拉斯算子都与模拟物理、生物和经济系统的布朗运动相互作用，甚至与更一般的Levy过程相互作用，Levy过程被广泛用于模拟各种金融系统，这些金融系统会发生更瞬时的变化或跳跃。 鞅、拉普拉斯算子和布朗运动之间相互作用的根源是一个深刻的数学理论，它使这些应用在科学上是合理的。 这个项目涉及其中几个核心问题，并将它们与其他几个数学分支联系起来，包括更一般的偏微分方程及其在热扩散、波传播及其与随机现象的联系方面的应用。这些项目将涉及研究生。 研究结果将通过在专业期刊、讲座和网络上发表的出版物传播。 将做出真诚的努力，让学生和年轻的博士接触（并参与）。从代表性不足的群体到这项研究，并增加他们对数学的参与。