Spectral asymptotics for stable processes

拟议的项目将调查位于概率，偏微分方程和谱理论的接口的几个数学问题。在概率方面，这些问题是制定的随机过程，以自然的方式布朗运动的行为。这些过程，被称为列维过程，以法国数学家保罗·列维命名，在数学的许多领域都扮演着重要的角色。 它们在生命科学、物理科学和经济学等不同领域都有应用。 它们在金融市场建模中特别有用。在偏微分方程和谱理论方面，它们涉及经典理论的扩展，用于模拟涉及分数扩散的现象。 数学上的问题如下。(1)具有光滑位势的旋转对称稳定过程（分数拉普拉斯）生成元的欧氏空间上薛定谔算子的迹渐近性。 在拉普拉斯算子的情况下，这产生了著名的热不变量的潜力，用于逆散射和其他问题的光谱理论所产生的数学物理领域。 散射和共振的稳定过程和连接的分数拉普拉斯算子的热不变量的问题将进行探讨。(2)欧氏空间上域的一个两项Weyl渐近律。 马克·卡茨在50年代早期引入的概率技术证明了著名的“外尔第一定律”关于拉普拉斯算子（布朗运动）的本征值随区域体积的增长，可以用来将这个结果推广到许多其他Levy过程的本征值。这些概率/热方程方法未能给出“外尔第二定律？这是一个更深层次的结果，并在1980年由Ivrii证明了拉普拉斯算子，此后不久由Melrose证明。 另一方面，Ivrii和Melrose的分析技术也失败了，主要是由于处理跳跃过程所需的边界条件。目前的概率和分析方法的失败使这些问题极具挑战性。为了在这些问题上取得进展，必须开发全新的技术。这些技术可能会影响数学中的几个领域，包括概率和谱理论。这些项目将涉及研究生的培训。 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。 与以前授予PI的NSF赠款一样，将努力让学生和年轻的博士参与进来。来自于研究中代表性不足的群体。

The proposed projects will investigate several mathematical problems which lie at the interface of probability, partial differential equations and spectral theory. On the probability side, the problems are formulated in terms of the behavior of stochastic processes which extend in a natural way Brownian motion. These processes, called Levy processes after the French mathematician Paul Levy, play an important role in many areas of mathematics. They have applications in fields as diverse as life and physical sciences and economics. They have been particularly useful in the modeling of financial markets. On the partial differential equations and spectral theory side, they involved extensions of classical theories to operators which are used to model phenomena involving fractional diffusions. Mathematically the problems are as follows. (1) Trace asymptotics for Schrodinger operators on Euclidean spaces for the generator of rotationally symmetric stable processes (fractional Laplacian) with smooth potentials. In the case of the Laplacian this produces the famous heat invariants for potentials which are used in inverse scattering and other problems in spectral theory arising in areas of mathematical physics. Questions of scattering and resonances for stable processes and connections to heat invariants for the fractional Laplacian will be explored. (2) A two-term Weyl's asymptotic law for domains on Euclidean spaces. The probabilistic techniques introduced by Mark Kac in the early 50's to prove the celebrated "Weyl's First Law" on the growth of the eigenvalues of the Laplacian (Brownian motion) in terms of the volume of the domain can be used to extend this result to the eigenvalues of many other Levy processes. These probabilistic/heat equation methods fail to give "Weyl's Second Law? which is a much deeper result and which was proved in 1980 for the Laplacian by Ivrii and shortly thereafter by Melrose. On the other hand, the analysis techniques of Ivrii and Melrose also fail primarily due to the boundary conditions needed to deal with processes with jumps. The failure of both the current probabilistic and analytic methods makes these problems extremely challenging. In order to make progress on these questions, brand new techniques will have to be developed. Such techniques will likely impact several fields in mathematics, including probability and spectral theory. These projects will involve the training of graduate students. The results will be disseminated through publications in professional journals, lectures and on the web. As in previous NSF grants awarded to the PI, serious efforts will be made to involve students and young Ph.D.'s from underrepresented groups on this research.