Interaction between areas of Mathematics related to internal structures of fractals

与分形内部结构相关的数学领域之间的相互作用

• 批准号: 23340025
• 负责人: KIGAMI Jun
• 金额: $ 11.9万
• 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: 2011
• 资助国家: 日本
• 起止时间: 2011-04-01 至 2014-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-23340025/
• 关键词: フラクタル; 解析学; 確率論; 力学系; ラプラシアン; 熱核

In this project, we have studied internal structure of fractal sets from the broad viewpoint of analysis, geometry and algebra. In particular, we have considered stochastic processes and potential theory on fractals, which are thought of as models of objects in the nature. For instance, we have obtained conditions to ensure the existence of scaling relation of space-time on the heat conductance on fractals. Moreover, we have studied equilibrium state of heat on spaces with infinitely many holes and obtained a condition for the equilibrium state to be smooth.

本课题从分析学、几何学和代数学的角度研究了分形集的内部结构。特别是，我们已经考虑了随机过程和潜在的分形理论，这被认为是自然界中的对象的模型。例如，我们得到了分形上热传导时空标度关系存在的条件。此外，我们还研究了具有无限多个洞的空间上的热平衡态，得到了平衡态光滑的一个条件。

项目成果

Dirichlet forms on a noncompact Cantor set and random walk on its defining tree

狄利克雷在非紧康托集上形成，并在其定义树上随机游走

• 作者: M. Jimbo;K. Shiromoto;Y. Lin;H. Nakada;Y Morita;T.Sugawa;Shun Shimomura;宍倉光広;木上 淳
• 通讯作者: 木上 淳

Boundary Harnack inequality for Markov processes with jumps

带跳跃的马尔可夫过程的边界 Harnack 不等式

• DOI: 10.1090/s0002-9947-2014-06127-8
• 发表时间: 2015
• 期刊: Trans. Amer. Math. Soc.
• 作者: K. Bogdan;T. Kumagai and M. Kwaśnicki
• 通讯作者: T. Kumagai and M. Kwaśnicki

On the equivalence of parabolic Harnack inequalities and heat kernel estimates

• DOI: 10.2969/jmsj/06441091
• 发表时间: 2012-10
• 期刊: Journal of The Mathematical Society of Japan
• 影响因子: 0.7
• 作者: M. Barlow;A. Grigor’yan;T. Kumagai

Geodesic Distances and Intrinsic Distances on Some Fractal Sets

• DOI: 10.4171/prims/129
• 发表时间: 2013-11
• 期刊: Publications of The Research Institute for Mathematical Sciences
• 影响因子: 1.2
• 作者: M. Hino

Completion of the Sierpinski gasket minus the unit interval

完成谢尔宾斯基垫片减去单位间隔

• 发表时间: 2013
• 作者: Akahira;M.;Jun Kigami
• 通讯作者: Jun Kigami