Mathematical Sciences: Brownian Paths and Potential Theory

主要研究人员将继续他对布朗运动的路径性质的研究，他将把他的知识和技术应用到几个领域：与布朗运动有关的分形模型，三维布朗路径上的调和测度，不相交问题，仅举几例。为了详细阐述第一个主题，曼德尔布洛特创造了“分形”一词来描述现实世界中非平稳现象的数学模型。他认为有一类分形族是从布朗运动轨迹中衍生出来的。分形学的理论仍处于早期阶段，研究分形学模型的最基本性质是研究该领域中最重要的研究任务之一，以了解这种模型是否以及在多大程度上是现实的。特别是，将估计几个分形图的Hausdorff维度。此外，还将进行计算机模拟，以求出几个分形集的Hausdorff维度。

The principal investigator will continue his study of path properties of Brownian motion, and he will apply his knowledge and techniques to several areas: fractal models related to Brownian motion, harmonic measure on 3-dimensional Brownian paths, non-intersection problems, to name only a few. To elaborate on the first topic, Mandelbrot coined the word "fractal" to describe mathematical models of nonsmooth real world phenomena. One family of fractal which he considered was derived from Brownian motion paths. The theory of fractals is still in its early stages and one of the most important research tasks in this area is to study the most elementary properties of fractal models in order to learn if and to which extent such models are realistic. In particular the Hausdorff dimensions of several fractals will be estimated. Moreover, computer simulations will be performed to find the Hausdorff dimensions of several fractal sets.