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CCMS Colloquium: Brownian Motion, Random Fractals, and Beyond (Yimin Xiao, Michigan State Uni)
March 26 @ 4:15 pm - 5:30 pm
Speaker: Yimin Xiao, Professor of Statistics and Probability, Michigan State University
Title: Brownian Motion, Random Fractals, and Beyond
Abstract: Brownian motion is not only a fascinating object in mathematics but also a powerful stochastic model that has been applied in many scientific fields, from physics to finance, to biological sciences. The sample function of Brownian motion is nowhere differentiable and generates various interesting fractal sets and measures. It is natural to apply tools from Fractal Geometry (e.g. Hausdorf dimension, packing dimension) to study the fine properties of Brownian motion. There is an enormous literature on sample path properties of Brownian motion, including many fractal and multifractal properties.
In this talk, we first provide an overview of some regularity and fractal properties of Brownian motion and its local times. These results are fundamental for the development of related theories for other stochastic processes. We then describe extensions of these results to Levy stable processes and fractional Brownian motion, which are two important relatives of Brownian motion in the areas of Markov processes and Gaussian processes, respectively.
Bio: Yimin Xiao is an MSU Research Foundation Professor in Statistics and Probability at Michigan State University. He received his Ph.D. from the Ohio State University in 1996. After completing his postdocs at the University of Utah and Microsoft Research, he joined Michigan State University in 2000. His research interests include random fields, Gaussian and Lévy processes, stochastic partial differential equations, extreme value theory, random fractals, and statistical analysis of spatial and spatio-temporal models. He has published about 160 articles in peer-reviewed journals.
Dr. Xiao has been a visiting professor at several universities in China, France, Germany, Hong Kong, Sweden, and Switzerland and has been invited as a speaker at various international conferences. He is a Fellow of the Institute of Mathematical Statistics.
