Stochastic analysis and its application to analysis of differential operators

随机分析及其在微分算子分析中的应用

基本信息

批准号：
15540116
负责人：
MATSUMOTO Hiroyuki
金额：
$ 2.24万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

The stochastic differential equation for the Brownian motion on the Poincare upper half plane (the hyperbolic plane), a diffusion process generated by a half of the Laplacian, is explicitly solved and we have an concrete representation for the Brownian motion as a Wiener functional. In this research, as an extension of this known fact, we showed that the horizontal life of the Brownian motion to the bundle of orthonormal frames also has an expression as a Wiener functional. Based on this representation, we may show a probabilistic representation for the heat kernel of the Laplacian acting on the differential forms and give a proof of the Selberg trace formula for the differential 1-forms on a compact Riemannian surface, which may be given as a quotient space of the upper half plane by a hyperbolic discrete subgroup of the isometry group. This is an analytic and/or geometric proof and we do not need the harmonic analysis. Moreover we have obtained the Selberg trace formula in a very exp … More licit form, since we have restrict ourselves to the two-dimensional case.The research for an extension to the general dimension case has been continued. In the two dimensional case, we can represent the rotation part of the horizontal lift by using an auxiliary one-dimensional Brownian motion and this representation plays a crucial role. We have not found a corresponding representation and this should be the next task. If we find such a representation, we will be able to give a proof for the Selberg trace formula following the idea of McKean which has been the basis of this research.When we apply probability theory to the analysis on the upper half plane, the exponential Wiener functionals which is an integral of a geometric Brownian motion appear. Some studies on these functionals has been continued since the Wiener functionals of the same type also appear in the theory of Mathematical Finance and a study for some diffusion processes in random environments. In a joint project with Professot YOR, who is a foreign co-worker in thie research, we gathered some results and applications of these exponential Wiener functionals and gave an insight from analytic point of view. The results have been published in a journal. Less

在繁殖力上半平面上的布朗运动的随机微分方程（双曲平面）是由Laplacian的一半产生的差异过程，明确解决了，我们具有布朗尼运动作为维也纳功能的混凝土表示。在这项研究中，作为这一已知事实的扩展，我们表明，布朗运动向正统框架束的水平寿命也表达为Wiener功能。基于此表示，我们可能会显示出作用于差异形式的拉普拉斯的热核的概率表示，并证明了Selberg Trace公式在紧凑的Riemannian表面上的差分1形式，可以通过均值依次依次的依次分组的差异平面给出，这可以作为上半平面的引号给出。这是一个分析和/或几何证明，我们不需要谐波分析。此外，我们已经以非常有载体的形式获得了Selberg Trace公式，因为我们将自己限制在二维案例中。延伸到一般维度案例的研究已经继续进行。在两个维度的情况下，我们可以使用辅助一维的布朗运动来表示水平升降的旋转部分，并且这种表示起着至关重要的作用。我们尚未找到相应的表示形式，这应该是下一个任务。如果我们找到了这样的表示形式，我们将能够根据McKean的想法提供Selberg Trace公式的证明，这是这项研究的基础。当我们将概率理论应用于上半平面的分析时，指数的Wiener功能是几何布朗尼运动的组成部分。由于同一类型的Wiener功能也出现在数学金融理论以及对随机环境中某些扩散过程的研究中也出现了一些有关这些功能的研究。在与Thie研究的外国同事的专业YOR联合项目中，我们收集了这些指数Wiener功能的一些结果和应用，并从分析的角度提供了见解。结果已发表在杂志上。较少的