FBM, Hypoelliptic Processes, and Path Integrals

这项提议主要涉及三个主题。一是研究分数布朗运动驱动的亚椭圆型随机微分方程解的密度光滑性。二是继续分析黎曼流形上Wiener测度的某些有限维几何逼近。最终目标是包括超对称设置，这可能需要在非对易概率空间中进行分析。P.I.预计该项目的这一方面将与随机矩阵理论相联系。第三个问题致力于证明热方程在某些无限维李群上的抛物正则性结果。以这样或那样的方式，这些问题中的每一个都可以被重塑(至少是启发式地)为涉及路径积分的问题。这里提出的许多项目都是由建立量子化杨-米尔场这一根本问题松散地推动的。关于这个问题的描述以及它对物理学中所谓的标准模型的重要性，请参阅克莱数学研究所关于量子化杨-米尔斯场的问题。分数布朗运动(FBM)是由A.N.Kolmogorov于1940年隐含地提出的。(科尔莫戈罗夫似乎想到了湍流理论的可能应用。)Mandelbrot和Van Ness(1968)后来的一篇论文描述了FBM的一些可能的应用，包括将其用于经济和水文模型。最近的文献包括FBM在神经网络建模、金融工具定价以及理解星系团方面的应用。这一建议的主要项目之一是研究由分数布朗运动驱动的这类动力系统的统计行为。在这项研究拨款期间，私人投资委员会还将专注于另外两个项目。其中一个项目是P.I.S计划的继续，以更好地理解费曼“路径积分”技术，该技术用于原子和亚原子粒子的量子力学描述。尽管费曼的路径积分得到了广泛的研究，但其数学基础充其量也是微不足道的。第三个项目是将前两个项目的结果概括到“无限维”。事实证明，对无限维度看似非常抽象的概括，正是理解自然界基本力的量子力学描述所需要的。(这个主题在粒子物理文献中的量子场论的标题下。)如何在数学上严格理解相互作用的量子场论是一个长期存在的重大挑战。这项提议有一个重要的研究生培训部分，因为许多问题将由P.I.S的学生来解决。

This proposal is primarily concerned with three topics. The first is to study the smoothness properties of densities for laws of solutions to hypoelliptic stochastic differential equations driven by fractional Brownian motion. The second is to continue to analyze certain finite dimensional geometric approximations to Wiener measure on a Riemannian manifold. The eventual goal is to include the super-symmetric setting which will likely entail analysis in non-commutative probability spaces. The P.I. expects this aspect of the project will make contact with the theory of random matrices. The third problem is devoted to proving parabolic regularity results for heat equations on certain infinite dimensional Lie groups including loop groups. In one way or another each of these problems may be recast (at least heuristically) as problem involving path integrals. Many of the projects proposed here are loosely motivated by the fundamental problem of constructing quantize Yang-Mills fields. See the Clay Mathematics Institute problem pertaining to quantized Yang - Mills fields for a description of this problem and its importance to the so called standard model in physics. Fractional Brownian motion (fBm) was implicitly introduced by A. N. Kolmogorov in 1940. (Kolmogorov seemed to have in mind possible applications to the theory of turbulence.) A later paper by Mandelbrot and Van Ness (1968) describes a number of possible applications for fBm including using it for economic and hydrology models. The recent literature includes applications of fBm in the modeling of neural networks, the pricing of financial instruments, and to understanding the clustering of galaxies. One of the main projects of this proposal is to study the statistical behavior of these types of dynamical systems which are driven by fractional Brownian motion. The P.I. will also focus on two other projects during the period of this research grant. One of these projects is a continuation of the P.I.'s program to better understand the Feynman "path integral" techniques which are used in the quantum mechanical description of atomic and sub-atomic particles. Although highly studied, the mathematical footing of Feynman's path integrals is still tenuous at best. The third project is to generalize the results of the first two projects to "infinite dimensions." It turns out that the seemingly very abstract generalization to infinite dimensions is precisely what is needed in order to understand the quantum mechanical description of the fundamental forces in nature. (This topic goes under the heading of quantum field theory in the particle physics literature.) It is a long standing major challenge to make mathematically rigorous sense of interacting quantum field theories. This proposal has a significant graduate training component as a number of the problems will be tackled by the P.I.'s students.