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Analysts on harmonic maps over geometric singular spaces via Dirichlet forms

通过狄利克雷形式分析几何奇异空间上的调和映射

基本信息

批准号：
16540201
负责人：
KUWAE Kazuhiro
金额：
$ 2.37万
依托单位：
Kumamoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540201/
关键词：
Dirichlet form harmonic map subharmonic function variational convergence Gromov-Hausdorff convergence Alexandrov space Kato class heat kennel アレキサンドロフ空間

项目摘要

We establish the following result :1) Variational convergence of metric measure spaces:We introduce a natural definition of Lp-convergence of maps. $pge 1$, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the Lp-convergence, we establish a theory of variational convergences. We prove that the Poincare inequality with some additional condition implies the asymptotic compactness. The asymptotic compactness is equivalent to the Gromov-Hausdorff compactness of the energy-sublevel sets. Supposing that the targets are CAT(0)-spaces, we study convergence of resolvents. As applications, we investigate the approximating energy functional over a measured metric space and convergence of energy functionals with a lower bound of Ricci curvature. This work was done with Prof. T. Shioya.2) Perturbation of symmetric Markov processes and its related stocha … More stic calculus:We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower orderperturbation of the L2-infinitesimal generator L of a general symmetric Markov process. Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an It^o formula for Dirichlet processes is obtained. This work was done with Professors Z.Q. Chen. P.J. Fitzsimmons and T.S. Zhang.3) Kato class measures over symmetric Markov processes :We show that $fin L^p(X ; m)$ implies $|f|dmin S_K^1$ for $p>D$ with $D>0$, where $S_K^1$ is a subfamily of Kato class measures relative to a semigroup kennel $p_t(x, y)$ of a Markov process associated with a (non-symmetric) Dirichlet form on $L^2(X ; m)$. We only assume that $p_t(x, y)$ satisfies the Nash type estimate of small time defending on $D$. No concrete expression of $p_t(X, V)$ is needed for the result. This wonk was done with M. Takahashi.4) Refinements of exceptional sets with respect to (n, p)-capacity oven symmetric Markov processes:We establish a one to one correspondence between a class of smooth measures in the (n, p)-sense and a class of positive continuous additive functionals admitting (n, p)-exceptional sets. This work was done with A. Sato.5) Liouville theorems for harmonic maps to convex spaces over Markov chains:We give a Liouville type theorem for harmonic maps from the space equipped with the harmonicity of functions in terms of conservative Markov chains to convex spaces admitting barycenters. No differentiable structures for the domain and the target are assumed. This work was done with prof. k.Th. Sturm.6) Laplacian comparison theorem on Alexandrov spaces :We consider a directionally restricted version of the Bishop-Gromov relative volume comparison as generalized notion of Ricci curvature bounded below for Alexandrov spaces. We prove a Laplacian comparison theorem for Alexandrov spaces under the condition. As an application we prove a topological splitting theorem. This work was done with Prof. T.Shioya. Less

1）度量测度空间的变分收敛：引入映射的Lp-收敛的一个自然定义。$pge 1$，当定义域是度量空间关于度量Gromov-Hausdorff拓扑的收敛序列，目标是度量空间关于度量Gromov-Hausdorff拓扑的收敛序列时.在Lp-收敛的基础上，建立了变分收敛理论。证明了Poincare不等式在附加条件下蕴含渐近紧性。渐近紧性等价于能量子水平集的Gromov-Hausdorff紧性。假设目标是CAT（0）-空间，我们研究了预解式的收敛性。作为应用，我们研究了度量空间上的能量泛函的逼近以及能量泛函在Ricci曲率下界下的收敛性。这项工作是与T教授一起完成的。Shioya.2）对称马氏过程及其相关随机过程的扰动 ...更多信息 stic演算：我们提出了一个路径空间积分表示的半群与二次型获得的低阶扰动的L2-无穷小生成元L的一般对称马尔可夫过程。利用时间反演，我们引入了对称马氏过程的零能可加泛函的一个随机积分，推广了S。中尾讨论了这类随机积分的各种性质，得到了Dirichlet过程的一个It^o公式.这项工作是与Z.Q.教授一起完成的。尘PJ菲茨西蒙斯和TS Zhang. 3）对称马氏过程上的Kato类测度：我们证明了$fin L^p（X ; m）$蕴含$|F| dmin S_K^1$，其中$p>D$，$D>0$，$S_K^1$是关于L^2（X ; m）$上（非对称）Dirichlet型的马氏过程的半群核$p_t（x，y）$的Kato类测度的一个子集.我们只假设$p_t（x，y）$满足在$D$上小时间防御的Nash型估计。$p_t（X，V）$的具体表达式是不需要的。这是和M一起做的。Takahashi.4）关于（n，p）-容腔对称Markov过程例外集的加细：我们建立了一类（n，p）-意义下的光滑测度与一类允许（n，p）-例外集的正连续可加泛函之间的一一对应。这项工作是用A. Sato.5）马氏链上凸空间调和映射的Liouville定理：我们给出了从具有保守马氏链函数调和性的空间到允许重心的凸空间的调和映射的Liouville型定理。没有可微结构的域和目标被假定。这项工作是与教授k.t. Sturm.6）Alexandrov空间上的拉普拉斯比较定理：我们考虑Bishop-Gromov相对体积比较的方向限制版本作为Alexandrov空间的Ricci曲率有界的广义概念。在此条件下，我们证明了Alexandrov空间的一个Laplacian比较定理。作为应用，我们证明了一个拓扑分裂定理。这项工作是与T. Shioya教授一起完成的。少