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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.

我们完成了半群支配和半群的交织性质的研究。半群支配表示|T^^→_tu|【小于或等于】T_t|u|对于半群T^^→_t和T_t成立。这里u应该是一个向量值函数。典型的例子是黎曼流形上的微分形式。发电机的特性是已知的。本文给出了平方域算子的一个充分条件。关键的假设是积极性和局部性。交织性质是：对于两个生成子L， L^^→，DL = L^^→D + r，给出了半群及其解的充分必要条件。我们的目标是重建。Bakry-Emery T_2理论，但我们成功地包含了边界条件下的扩散过程，并给出了Bakry-Emery T_2理论的推广。在实际应用中，我们可以利用它们来证明Littlewood-Paley不等式和L^p乘数定理。实际上，我们已经考虑了带有边界的黎曼流形上的布朗运动，并在边界的第二种基本形式为正的假设下给出了它的Littlewood-Paley不等式。利用这一点，我们可以证明Riesz变换的L^p有界性。L^p乘子定理是为了给出φ和A上φ(A)是L_p上有界算子的充分条件。对于φ为拉普拉斯变换型的对称扩散过程的发生器，Stein证明了这一点。我们已经证明了同样的结果适用于黎曼流形上的Hodge-Kodaira算子。