Brownian motion on a random recursive Sierpinski gasket

• 成果类型: Article
• 署名作者: Hambly, BM
• 署名单位: University of Edinburgh
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 1059-1102
• 关键词: transition densities; nested fractals; carpet; sets

摘要

We introduce a random recursive fractal based on the Sierpinski gasket and construct a diffusion upon the fractal via a Dirichlet form. This form and its symmetrizing measure are determined by the electrical resistance of the fractal. The effective resistance provides a metric with which to discuss the properties of the fractal and the diffusion. The main result is to obtain uniform upper and lower bounds for the transition density of the Brownian motion on the fractal in terms of this metric. The bounds are not tight as there are logarithmic corrections due to the randomness in the environment, and the behavior of the shortest paths in the effective resistance metric is not well understood. The results are deduced from the study of a suitable general branching process.