Integrated research of Probability Theory

概率论综合研究

基本信息

批准号：
14204008
负责人：
SHIGEKAWA Ichiro
金额：
$ 25.63万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

项目摘要

The main research area of the head investigator is diffusions in infinite dimension. At the same time, a research of diffusions on a Riemannian manifold is accomplished because geometric point of view is important in our research. We gave a probabilistic proof of the Littlewood-Paley inequality, the L^P norm equivalence between the gradient and the square of the generator, essential self-adjointness of a Schrodinger operator, and the spectral gap. The essential matter we used is the intertwining property between gradient and the generator. The use of Functional Analysis is crucial because it is irrelevant of the geometry of the space. In our general framework, we assume the logarithmic Sobolev inequality and the exponential integrability of the remaining term of the intertwining property. We also discussed the similar problem in a setting of Riemannian manifold with convex boundary.Further we considered a Schrodinger operator on the Wiener space of the form L+V,L beging the Ornstein-Uh … More lenbeck operator. We gave a characterization of the generator domain and proved the essential self-adjointness and the spectral gap of the Schrodinger operator under a suitable condition of the potiential V.We also showed the Littlewood-Paley inequality for the the Schrodinger operator. Since we have a potential term, we need a modification of the standard proof. This method works for a Hodge-Kodaira operator on a Riemannian manifold with a potential.In this project, we have held several symposiums and gave financial support for participants. One of them is "Stochastic Analysis and related fields" that was a research project of the Research Institute of Mathematical Sciences in 2002. We invited Professors McKean, Ustunel, Rockner, etc., from abroad. The others are "Probability Summer School" in 2003,2004. We gave introductory lectures of frontier of recent research for graduated students. We could accomplish stimulating discussions. We also held every year "Stochastic Processes and related fields", which gave fruitful communication between researchers in Japan. As a sum, we held 24 symposiums during three years and invited 14 foreign researchers and produced many results. Less

首席研究员的主要研究领域是无限维度的差异。同时，由于几何学的观点在我们的研究中很重要，因此完成了关于黎曼流派差异的研究。我们给出了利特伍德 - 帕利不平等的概率证明，发电机的梯度和平方之间的l^p规范等效性，schrodinger操作员的基本自我相关性以及频谱间隙。我们使用的必需材料是梯度和发电机之间的交织属性。功能分析的使用至关重要，因为它与空间的几何形状无关。在我们的一般框架中，我们假设对数Sobolev的不等式和相互交织属性的其余项的指数整合。我们还讨论了在带有凸边界的Riemannian歧管的环境中的类似问题。我们考虑了l+v的Wiener空间上的Schrodinger操作员，bing着Ornstein-uh…更多的Lenbeck操作员。我们给出了发电机结构域的特征，并证明了在电势V的适当条件下，Schrodinger操作员的基本自我相关性和光谱间隙。我们还显示了Schrodinger操作员的Littlewood-Paley不平等。由于我们有一个潜在的术语，因此我们需要修改标准证明。该方法适用于具有潜力的Riemannian歧管上的Hodge-Kodaira运营商。在该项目中，我们举办了几个研讨会，并为参与者提供了财政支持。其中之一是“随机分析和相关领域”，它是2002年数学科学研究所的研究项目。我们邀请了来自国外的McKean，Ustunel，Rockner等教授。其他是2003年，2004年的“概率暑期学校”。我们为毕业生的最新研究介绍了介绍。我们可以完成刺激的讨论。我们每年还举办“随机过程和相关领域”，这在日本研究人员之间进行了富有成果的沟通。总而言之，我们在三年内举行了24次研讨会，并邀请了14位外国研究人员，并产生了许多结果。较少的