Dirichlet space and analysis of harmonic map over the space of Gromov-Hausdorff limit spaces

狄利克雷空间与格罗莫夫-豪斯多夫极限空间上的调和映射分析

基本信息

批准号：
13640220
负责人：
KUWAE Kazuhiro
金额：
$ 2.18万
依托单位：
Yokohama City University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640220/
关键词：
Dirichlet form Harmonic map Variational convergence Asymtotic relation Gamma convergence Mosco convergence CAT(0)-space Gromov-Hausdorff convergence Alexandrov spaces Calabi s strong maximum principle 加藤クラスハーディクラスアレキサンドロフ空間ドリフ

项目摘要

(1)The study of variational convergence over metric measured space : This result is a joint work with Professor Takashi Shioya, who is an associate professor of Graduate School of Mathematical Institute, Tohoku University. We introduce a notion called asymptotic relation over a direct sum of metric spaces, which includes the notion of Gromov-Hausdorff convergence as an example. We discuss several notions of functionals over it, for example, Gamma convergence, Mosco convergence and compact convergence and so on. We give a sufficient condition for the compact convergence of functionals. We also prove a sufficient condition for the convergence of resolvents of energy functionals over CAT(0)-spaces.(2)The study on the stochastic representation of semigroups obtained from a non-symmetric perturbation : This study is a joint work with Professors P.J.Fitzsimmons, Z.Q.Chen and T.S.Zhang. Consider a symmetric regular Dirichlet form and the associated Hunt process admitting jumps of its sample paths. We consider a non-symmetric perturbation by use of locally square integrable martingale additive functionals and a continuous additive functional of finite variation. We prove that the corresponding semigroup has a stochastic representation in terms of time reverse operator on sample paths.(3)The study of Calabis type strong maximum principle : We give a stochastic proof of an extension of E.Calabi's strong maximum principle in the framework of strong Feller diffusion processes associated with local regular semi-Dirichlet forms. Our results can be applicable to the Gromov-Hausedorff limit space over a family of compact Riemannian manifolds with uniform lower bounds of Ricci curvature and uniform bounds of diameter.

（1）对公制测量空间的变异收敛研究：这是与Tohoku大学数学研究所副教授Takashi Shioya教授的联合工作。我们在直接的度量空间中引入了一个称为渐近关系的概念，其中包括Gromov-Hausdorff收敛的概念。我们讨论了几个功能上的概念，例如伽马收敛，MOSCO收敛和紧凑收敛等。我们为功能的紧凑收敛提供了足够的条件。我们还证明了能量功能在CAT（0）空间上的分解的足够条件。（2）（2）对从非对称扰动获得的半群的随机表示研究的研究：这项研究是与P.J.教授的共同工作。 Fitzsimmons，Z.Q.Chen和T.S.Zhang。考虑一种对称的常规差异形式和相关的狩猎过程，该过程承认其样品路径的跳跃。我们通过使用局部可集成的Martingale添加剂功能和有限变化的连续添加功能来考虑非对称扰动。我们证明，相应的半群在样本路径上的时间反向运算符方面具有随机表示。（3）卡拉比斯类型的研究强度最大原理：我们给出了E.Calabi在框架中强大最大原理扩展的随机证明与局部常规半二级联形式相关的强伐木扩散过程的强大扩散过程。我们的结果可以适用于紧凑型riemannian歧管的Gromov-Hausedorff限制空间，其直径均匀的下限和直径均匀的界限。