Integrated research of Probability Theory

首席研究员的主要研究领域是无限维中的扩散。同时，由于几何的观点在我们的研究中起着重要的作用，我们完成了对黎曼流形上扩散的研究。我们给出了Littlewood-Paley不等式、生成元的梯度与平方之间的L^P范数等价性、Schrodinger算子的本质自伴性以及谱间隙的概率证明。我们使用的本质是梯度和生成元之间的交织性质。功能分析的使用是至关重要的，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不等式和指数可积性的纠缠性质的剩余项。我们还讨论了在具有凸边界的黎曼流形上的类似问题，并进一步讨论了Wiener空间上L+V形式的Schrodinger算子，L开始于Ornstein-Uh ...更多信息 Lenbeck算子给出了生成域的一个刻划，证明了在势V的适当条件下，Schrodinger算子的本质自伴性和谱间隙，并证明了Schrodinger算子的Littlewood-Paley不等式.因为我们有一个潜在的条款，我们需要一个标准的证明修改。该方法适用于具有势函数的黎曼流形上的Hodge-Kodaira算子。在这个项目中，我们已经举办了几次研讨会，并为参与者提供了资金支持。其中之一是2002年数学科学研究所的研究项目“随机分析及相关领域”。我们邀请了麦基恩、乌斯图内尔、洛克纳等教授，从国外其他的是2003年和2004年的“概率暑期学校”。我们为毕业生提供了最新研究前沿的介绍性讲座。我们可以进行有启发性的讨论。我们还每年举办“随机过程和相关领域”，这使日本研究人员之间进行了富有成效的交流。三年来，我们共举办了24次学术研讨会，邀请了14位国外研究人员，取得了许多成果。少

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