Levy Processes, Martingales and Spectral Theory

这个项目涉及概率、调和分析和拉普拉斯、分数拉普拉斯及其相对论形式的谱理论之间的几个问题和猜想。它的目的是通过Levy-Khintchine公式对Levy过程进行泛函修改，得到一类傅立叶乘子的新的尖锐不等式。给出了布朗运动半群、稳定过程和相对论稳定过程的迹的普适界，并研究了这些过程的谱计数函数的二阶渐近性。这些调查的统一主题是使用基本的利维过程的符号。由分数拉普拉斯波动方程理论发展而来的鞅理论、热核估计和技巧(甚至可能尚未被发现)将在这些研究中发挥关键作用。一百多年前发明的用于解释公平博弈理论的鞅在数学的许多分支中扮演着基础角色，它们在物理、生物和社会科学的无数领域中都有应用。它们与一种称为拉普拉斯算子的二阶微分算子密切相关。这个算符是以200多年前的数学天文学家皮埃尔-西蒙·德·拉普拉斯侯爵的名字命名的，它为热、波、电、磁和流体理论提供了数学基础。它不仅与模拟物理、生物和经济系统的布朗运动相互作用，而且与更一般的Levy过程相互作用，后者被广泛用于对各种具有更多瞬时变化或跳跃的金融系统建模。在令、拉普拉斯算子和布朗运动之间相互作用的根源是一种深刻的数学理论，它使这些应用变得科学可靠。这个项目涉及这些核心问题中的几个，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的扩散、波的传播以及它们与随机现象的联系中的应用。这些项目将涉及研究生。结果将通过专业期刊、讲座和网络上的出版物传播。将做出真诚的努力，让学生和年轻的博士S从代表不足的群体中接触到这项研究，并增加他们对数学的参与。

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.