Multilateral research of stochastic analysis in infinite dimensional spaces

无限维空间随机分析的多边研究

基本信息

批准号：
11440045
负责人：
SHIGEKAWA Ichiro
金额：
$ 3.52万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

项目摘要

We have accomplished the research of the semigroup domination and the intertwining property of semigroups. The semigroup domination stands for that |T^^→_tu|【less than or equal】T_t|u| holds for semigroup T^^→_t and T_t. Here u are supposed to be a vector valued function. Typical example is a differential form on a Riemannian manifold. The characterization in terms of the generator is known. We give here a sufficient condition in terms of square field operator. Crucial assumptions are the positivity and the locality. Intertwining property is the following property: for two generator L, L^^→, it holds that DL = L^^→D + R. We give necessary and sufficient conditions in terms of the semigroup and the resolvents. Our aim has been to reconstruct the.Bakry-Emery T_2 theory but we have succeeded in including diffusion processes with boundary condition and it gives a generalization of Bakry-Emery T_2 theory.For application, we can make use of them to show the Littlewood-Paley inequality and the L^p multiplier theorem. In fact, we have considered the Brownian motion on an Riemannian manifold with boundary and show the Littlewood-Paley inequality for it under the assumption of positivity of the second fundamental form of the boundary. Making use of this, we can show the L^p boundedness of the Riesz transformation. L^p multiplier theorem is to give an sufficient condition on φ and A so that φ(A) is a bounded operator on L_p. Stein showed this for a generator of a symmetric diffusion process with φ being of Laplace transform type. We have shown that the same result holds for the Hodge-Kodaira operator on a Riemannian manifold.

我们已经完成了半群统治和半群的交织特性的研究。 Semigroup支配代表| T ^^→_tu | |小于或等于】T_T| U |保留Semigroup t ^^→_t和T_T。在这里，U预计将是矢量有价值的函数。典型的例子是riemannian歧管上的差异形式。以发电机的形式进行的表征是已知的。我们在此处给予方形野外操作员的条件足够。关键的假设是积极性和所在地。交织属性是以下属性：对于两个发电机l，l ^^→，它认为dl = l ^^→d + R.我们根据半群和分辨率提供了必要的足够条件。我们的目的是重建bakry-Emery T_2理论，但我们成功地包括了具有边界条件的差异过程，它给出了Bakry-Emery T_2理论的概括。对于应用，我们可以使用它们来显示Littlewood-Paley的不平等和L^P Multipleer定理。实际上，我们考虑了带边界的Riemannian歧管上的布朗运动，并在边界的第二种基本形式的阳性假设下显示了Littlewood-Paley的不平等。利用这一点，我们可以显示Riesz转换的L^p界限。 l^p乘数定理是在φ上给出足够的条件，因此φ（a）是l_p上的有限算子。斯坦（Stein）为对称扩散过程的发电机展示了这一点，其φ是拉普拉斯变换类型的生成器。我们已经表明，在Riemannian歧管上，Hodge-Kodaira操作员的结果相同。