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Research on vibrations and diffusions on fractals

分形振动和扩散研究

基本信息

批准号：
11837008
负责人：
KIGAMI Jun
金额：
$ 2.3万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11837008/
关键词：
fractals self-similar sets Laplacian diffusion Dirichlet form heat kernel asymptotic behavior 熱方程式自己相似

项目摘要

In this research project, we obtained the following five main results related with analysis on fractals.(1) Markov property of Dirichlet forms on self-similar setsWe showed the Markov property of Kusuoka-Zhou's Dirichlet forms on self-similar sets.(2) Self-similarity of volume measures associated with Laplacians on p.c.f. self-similar fractalsWe obtained a sufficient condition for the self-similarity of the volume measure, which is defined by using operator theoretic trace. We also showed that the sufficient condition holds in the case of standard Laplacian on the Sierpinski gasket.(3) Green's function on fractalsWe obtained an algorithm to calculate the diagonal of Green's function and used the algorithm to investigate the maximum value of Green's function.(4) Large deviations for Brownian motion on the Sierpinski gasketWe showed that Varadhan type estimate and Schilder type Large deviation do not hold for the case of the standard Laplacian on the Sierpinski gasket(5) Multifractal formalisms for the local spectral and walk dimensionsWe showed the multifractal nature of the local spectral and walk dimensions associated with the Laplacians on the self-similar sets.

在本研究项目中，我们获得了以下五个与分形分析相关的主要成果。（1）自相似集上狄利克雷形式的马尔可夫性质我们展示了自相似集上Kusuoka-Zhou狄利克雷形式的马尔可夫性质。（2）与p.c.f.上的拉普拉斯算子相关的体积测度的自相似性。自相似分形我们得到了体积测度自相似的充分条件，它是用算子理论迹定义的。我们还证明了在谢尔宾斯基垫片上标准拉普拉斯算子的情况下充分条件成立。(3)分形上的格林函数我们得到了计算格林函数对角线的算法，并用该算法研究了格林函数的最大值。(4)谢尔宾斯基垫片上布朗运动的大偏差我们证明了Varadhan型估计和Schilder型大偏差对于谢尔宾斯基垫片上的标准拉普拉斯算子的情况不成立。(5)局部谱和游走维数的多重分形形式我们展示了与自相似集上的拉普拉斯算子相关的局部谱和游走维数的多重分形性质。