Advanced topics in probability

The course will go through a number of topics in advanced probability. While the topics of this course will change for different years, for the upcoming course (spring 2025) the subject will be Brownian motion and to a lesser extent Hausdorff dimension. The course will contain many of the following topics: • The existence of Brownian motion. • Some of its sample path properties such as the fact that it is almost surely nowhere differentiable. • Definition of Hausdorff dimension and capacitarian dimension. • The structure of the zero set of 1-d Brownian motion (it is a perfect set of Hausdorff dimension 1/2). • The law of the Iterated Logarithm. • Arc sine laws. • The exact passage time distribution. • The Hausdorff dimension of Brownian motion in dimensions at least 2 is two and, more generally, the "dimension doubling" property. • The solution of the classical Dirichlet problem in terms of Brownian motion. • The existence of "double points" in 3 dimensions and their nonexistence in 5 dimensions. • The critical 4-dimensional case will be described and perhaps be proved. • Description of the first passage times process in terms of Poisson random measures.