某些随机分形集的一致维数及Assouad维数

自从Levy 和Taylor 在上世纪五十年代首先研究了Brownian运动和Levy过程的分形性质以来，随机集的分形性质研究成为随机场（过程）的一块重要理论，已有大量文献研究随机集的分形维数。但对很多复杂的随机集，通常的维数结果提供不了有效信息。为此，Kaufman引进了一致分形维数这一概念。随机集的一致分形维数能更全面深入地反映相应随机场（过程）的结构和几何性质。本项目拟通过对一些随机场（过程）像集、图集的一致维数（包括Fourier维数、Hausdorff维数、填充维数）及与之相关问题的系统研究，更好地探索和了解相应随机场（过程）的几何结构。本项目是随机场（过程）、分形理论及测度Fourier分析的交叉研究。预期建立一些重要随机集的一致分形维数结果，得到一些（随机）集合的修正Assouad维数, 探索修正Assouad维数与一致分形维数之间更深入的联系。

Fractal properties of random sets have become significant part of the theory on random fields (processes) since the pioneering works on Brownian motion and levy processes by Levy and Taylor. There is a huge literature on the fractal dimensions of random sets. But the usual dimensional results cannot provide effective information for some complicated random sets. Therefore, Kaufman introduced the so called uniform dimensions. The uniform dimension results can profoundly and comprehensively reflect the constructions and geometry properties of the corresponding random fields (processes). Our projection plans to explore the geometric constructions of some random fields (processes) by studying the uniform fractal dimensions (including Fourier dimension, Hausdorff dimension and packing dimension) of their image and graph sets and the related problems. This a research topic overlapped by random fields (processes), fractal theory and Fourier analysis of measures. We expect to establish the uniform fractal dimension results of some important random sets, obtain the modified Assouad dimensions of some (random) sets and explore the deeper relationships between modified Assouad dimensions and uniform fractal dimensions.