Research on vibrations and diffusions on fractals

点击翻译按钮获取中文摘要

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.