Speaker: Pro.Nanrui Xie（National Taiwan University）

Title: Lectures on Brownian Motions Fractal Dimensions

Time: Thus, Mar.7, 2019, PM:2:30-5:00

Location: Room 4224, Building No.4, Wushan Campus

Abstract:

We begin with the mathematical definition and the stochastic properties of BMs in R^d; d >=1. Then, we discuss two BM path properties; one is common for all d, and one is sharply different for different d = 1; 2; 3. Then, we discuss four random sets defined by a BM path and their fractal dimensions. The goal is to attract the attention to the planar BM paths, of which completely new fractal dimensions results, among others, lead two Fields Medals 2006 and 2010. The references:

        1. J.P. Kahane: Some Random Series of Functions, 2nd Edition. CUP 1985.

        2. K. Falconer: Foundation on Fractal Geometry. Wiley 1987.

        3. G.F. Lawler: Conformally Invariant Processes in the Plane. AMS 2012.

        4. P. Morters and Y. Peres: Brownian Motion. CUP 2010.

        5. P. Morters: Sample path properties of BM. Berlin Math LNs, 2011.

        6. N.-R. Shieh: several published papers related to sample-paths properties.