Mathematical Sciences: Brownian Paths and Potential Theory

首席研究员将继续研究布朗运动的路径特性，并将他的知识和技术应用于几个领域：与布朗运动相关的分形模型，三维布朗路径的谐波测量，非相交问题，仅举几例。为了详细阐述第一个主题，曼德布洛特创造了“分形”这个词来描述非光滑现实世界现象的数学模型。他认为分形的一类是由布朗运动路径推导出来的。分形理论还处于起步阶段，该领域最重要的研究任务之一是研究分形模型的最基本性质，以了解这些模型是否真实以及在多大程度上真实。特别是几个分形的豪斯多夫维数将被估计。此外，将进行计算机模拟，以找出几个分形集的豪斯多夫维数。

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.