Heat Kernel Analysis

本研究致力于有限维和无限维环境中涉及热方程的问题。需要解决以下问题。1.有限维亚椭圆热方程微分型不等式的存在性，包括对数Sobolev和Poincare（或质量间隙）型不等式的存在性. 2.黎曼流形的路空间和圈空间上与热核测度和Wiener测度有关的微分不等式的存在性3.紧黎曼流形的Loop空间上热核测度与钉住Wiener测度之间的关系。4.从有限维群到无限维循环群的类Fock空间表示的推广。5.将某些有限维近似推广到维纳测量，以包括物理学家使用的“超对称量子力学”设置。 在这项研究中要研究的大多数问题的动机，或者是来自研究一些领域的问题，即弯曲金属板中的热传播，量子力学，量子场论和概率论的某些方面。这些看起来完全不同的领域有着共同的数学描述，即“抛物型偏微分方程”或等价的“布朗运动”的概率理论。“在大多数有趣的（物理相关的）情况下，很少有可能找到P.I.研究的复杂偏微分方程的显式解。然而，我们常常可以发现解的有趣的和相关的性质。抛物型偏微分方程的一个典型现象是，在等待足够长的时间后，它们的解趋于稳态值。例如，如果金属板以不均匀的方式加热，然后单独放置，板中的热量将随着时间的推移重新分布，使得温度在整个板中达到恒定的平衡值。一个基本的问题，相关的“庞加莱”和“对数Sobolev”不等式，是如何快速做的抛物型偏微分方程所描述的系统在本研究中所研究的收敛到他们的平衡值的问题。在描述相对论性（即以接近光速运动）基本粒子的量子场论中，同样的收敛速度问题有另一种解释。对于这些理论来说，庞加莱不等式的存在有一种可取的解释，即数学理论并没有预言不合理的过多的基本粒子。抛物和相关的量子力学薛定谔方程的粒子解释将是另一个研究点。粒子的解释属于第一段提到的福克空间表示的标题。

This research is devoted to questions involving heat equations in both finite and infinite dimensional contexts. The following issues are to be addressed. 1. The existence of differential type inequalities related to hypoelliptic heat equations in finite dimensions, including the existence of logarithmic Sobolev and Poincare (or mass gap) type inequalities. 2. The existence of differential inequalities related to heat kernel measure and Wiener measures on path and loop spaces of Riemannian manifolds 3. The relationship between heat kernel measures and pinned Wiener measures on Loop spaces of compact Riemannian manifolds. 4. Extensions of Fock space like representations from finite dimensional groups to infinite dimensional loop groups. 5. The generalization of certain finite dimensional approximations to Wiener measures to include the "super symmetric quantum mechanics" setting used by physicists. Most of the questions to be studied in this research are motivated by, or are outgrowths of, questions coming from studying a number of fields, namely the spread of heat in curved metal plates, quantum mechanics, quantum field theories, and certain aspects of probability theory. These seemingly disparate fields turn out to share a common mathematical description, namely "parabolic partial differential equations" or equivalently the probabilistic theory of "Brownian motion." In most interesting (physically relevant) cases it is seldom possible to find explicit solutions to the complicated partial differential equations to be studied by the P.I. Nevertheless, it is often possible to discover interesting and relevant properties of the solutions. A typical phenomenon of parabolic partial differential equations is the fact that their solutions tend towards steady state values after waiting a sufficiently long time. For example, if a metal plate is heated in a non-uniform way and then left alone, the heat in the plate will redistribute itself over time so that the temperature reaches a constant equilibrium value throughout the plate. A fundamental problem, related to "Poincare" and "logarithmic Sobolev" inequalities, is the questions of how quickly do the systems described by the parabolic partial differential equations to be studied in this research converge to their equilibrium value. This same rate of convergence question has another interpretation in the context of quantum field theories describing relativistic (i.e. moving near the speed of light) elementary particles. For these theories the existence of a Poincare inequality has the desirable interpretation that the mathematical theory does not predict an unreasonable plethora of elementary particles. The particle interpretation of parabolic and related quantum mechanical Shrodinger equations will be another point of study. The particle interpretation goes under the title of Fock space representations which were mentioned in the first paragraph.