Mathematical Sciences: Brownian Paths and Potential Theory

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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.