Research on vibrations and diffusions on fractals

在本研究计画中，我们取得以下五个与分形分析相关的主要结果。(1)本文证明了Kusuoka-Zhou的Dirichlet型在自相似集上的马氏性。(2)p.c.f.上Laplacian体积测度的自相似性自相似分形利用算子理论迹定义了体积测度的自相似性，得到了体积测度自相似的一个充分条件。我们还证明了在Sierpinski垫片上的标准Laplacian的情况下，充分条件成立。(3)分形上的绿色函数给出了计算绿色函数对角线的一种算法，并利用该算法研究了绿色函数的最大值。(4)Sierpinski垫片上布朗运动的大偏差我们证明了Varadhan型估计和Schilder型大偏差不适用于Sierpinski垫片上的标准Laplacian的情形（5）局部谱维数和行走维数的多重分形形式我们证明了自相似集上Laplacian的局部谱维数和行走维数的多重分形性质。

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.