Spectral asymptotics for stable processes

拟议的项目将研究几个数学问题，这些问题存在于概率、偏微分方程组和谱理论的交界处。在概率方面，问题是根据随机过程的行为来表示的，随机过程以自然的布朗运动的方式扩展。这些过程被称为利维过程，以法国数学家保罗·利维的名字命名，在数学的许多领域发挥着重要作用。它们在生命、物理科学和经济学等领域都有应用。它们在金融市场的建模中尤其有用。在偏微分方程组和谱理论方面，它们涉及到经典理论对算子的扩展，这些算子被用来模拟涉及分数扩散的现象。从数学上讲，这些问题如下。(1)具有光滑势的旋转对称稳定过程(分数拉普拉斯)的生成元在欧氏空间上薛定谔算子的迹渐近性。在拉普拉斯的情况下，这产生了著名的势的热不变量，用于逆散射和数学物理领域中出现的光谱理论中的其他问题。我们将探讨稳定过程的散射和共振问题，以及分数拉普拉斯热不变量的联系。(2)欧氏空间上区域的二项Weyl渐近律。50年代初，S用概率论方法证明了著名的关于拉普拉斯(布朗运动)本征值随区域体积增长的“韦尔第一定律”，并将这一结果推广到许多其他Levy过程的本征值.这些概率/热方程方法不能给出“韦尔第二定律？这是一个更深层次的结果，伊夫里和梅尔罗斯在1980年证明了拉普拉斯定理。另一方面，伊夫里和梅尔罗斯的分析技术也失败了，主要是因为处理带有跳跃的过程所需的边界条件。当前概率方法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。这样的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。结果将通过专业期刊、讲座和网络上的出版物传播。与之前授予PI的国家科学基金会拨款一样，将认真努力让学生和年轻的博士生S参与这项研究。

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.