Favorite Points, Cover Times and Fractals

Abstract

In this course we follow recent advances in the study of the fractal nature of certain random sets, emphasizing the methods used to obtain such results. We focus on some of the fine properties of the sample path of the most basic stochastic processes such as simple random walks, Brownian motion, and symmetric stable processes. As we shall see, probability on trees inspires many of our proofs, with trees used to model the relevant correlation structure. Along the way we also mention quite a few challenging open research problems. Among the methods that will be detailed here are