Advanced topics in probability

About this programme

No fees are charged for EU and EEA citizens, Swedish residence permit holders and exchange students. 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 for Brownian motion Arc sine laws for Brownian motion The exact passage time distribution for Brownian motion The Hausdorff dimension of Brownian motion. 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 for Brownian motion The critical 4-dimensional case will be described and perhaps be proved Description of the first passage times process for Brownian motion in terms of Poisson random measures The course is given every other year, in the spring semester in odd-numbered years. Knowledge corresponding to the courses MSG110 Probability theory and MMA110 Integration theory. All eligible applicants who have applied before the deadline will be granted a place.