Probabilistic approach to analysis and geometry on metric measure spaces

度量测度空间上的分析和几何的概率方法

• 批准号: 22340036
• 负责人: KUWAE Kazuhiro
• 金额: $ 4.74万
• 依托单位: Kumamoto University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2010
• 资助国家: 日本
• 起止时间: 2010-04-01 至 2014-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-22340036/
• 关键词: ディリクレ形式; 調和写像; 劣調和関数; 変分収束; グロモフ・ハウスドルフ収束; アレキサンドロフ空間; 加藤クラス; 熱核

We obtained a Jensen's inequality over complete p-uniformly convex spaces. Further we proved the unique existence of the (non-linear) resolvent associated to a coercive proper lower semi continuous function satisfying a weaker notion of p-uniform convexity on a complete metric space and establish the existence of the minimizer of such functions as the large time limit of the non-linear resolvents, which generalizes the pioneering work by J. Jost for harmonic maps into CAT(0)-spaces. The results can be also applied to Lp-Wasserstein space over complete separable p-uniformly convex spaces. As an application, we solve an initial boundary value problem for p-harmonic maps into CAT(0)-spaces in terms of Cheeger type p-Sobolev spaces. On the other hand, we investigated the spectral bounds for symmetric Markov chains with positive n-step coarse Ricci curvature for not only functions but also maps into complete separable 2-uniformly convex spaces with geometric conditions.

在完备p-一致凸空间上得到了一个詹森不等式.进一步证明了完备度量空间上满足较弱p一致凸性概念的强制真下半连续函数的（非线性）预解式的唯一存在性，并建立了此类函数的极小元的存在性作为非线性预解式的大时间极限，从而将J. Jost关于调和映射的开创性工作推广到CAT（0）-空间. 这些结果也适用于完备可分p-一致凸空间上的Lp-Wasserstein空间。作为应用，我们利用Cheeger型p-Sobolev空间解决了CAT（0）-空间中p-调和映射的初边值问题.另一方面，我们研究了具有正n步粗Ricci曲率的对称马氏链的谱界，不仅对函数，而且对映射到具有几何条件的完全可分2-一致凸空间。

项目成果

On Calabi's strong maximum principle via local semi-Dirichlet forms

通过局部半狄利克雷形式论卡拉比的强极大值原理

• DOI: 10.1007/s11118-011-9266-5
• 发表时间: 2012
• 期刊: Potential Anal
• 影响因子: 1.1
• 作者: Y. Hirata;N. Kemoto and Y. Yajima;Teruya Minamoto;桑江一洋
• 通讯作者: 桑江一洋

Heat Flow on Alexandrov Spaces

• DOI: 10.1002/cpa.21431
• 发表时间: 2010-08
• 期刊: Communications on Pure and Applied Mathematics
• 影响因子: 3
• 作者: N. Gigli;Kazumasa Kuwada;Shin-ichi Ohta

Optimal transport and coupled diffusion by reflection

通过反射实现最佳传输和耦合扩散

• 发表时间: 2011
• 作者: Y. Mizumura;J. Kushida;K. Nishijima;K. Kodani;et al;桑田和正
• 通讯作者: 桑田和正

Concentration, Ricci curvature, and eigenvalues of Laplacian metric geometry and applications

拉普拉斯度量几何的浓度、里奇曲率和特征值及其应用

• 发表时间: 2011
• 作者: Mikihiko Nakao;Soh. Y. Suzuki;Ryosuke Itoh;Satoru Yamada;Takeo Higuchi;Tomoyuki Konno;T.Nakamura;Fumiyoshi Kajino;W. Deng;大石理子;Tadafumi Kishimoto;Y. Kawasaki;Junji Haba;T.Shioya
• 通讯作者: T.Shioya

Markov chains with OUivier's coarse Ricci curvature and harmonic maps into convex spaces

具有 OUivier 粗 Ricci 曲率和调和映射到凸空间的马尔可夫链

• 发表时间: 2012
• 作者: I. Ogawa;T Kishimoto;S Umehara他;K.Kuwae
• 通讯作者: K.Kuwae