Advanced topics in probability
About
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
Prerequisites and selection
Entry requirements
Knowledge corresponding to the courses MSG110 Probability theory and MMA110 Integration theory.
Selection
All eligible applicants who have applied before the deadline will be granted a place.
Facilities
Mathematical Sciences is a joint department of Chalmers/University of Gothenburg. Your education takes place in the spacious and bright premises of Mathematical Sciences at the Chalmers campus Johanneberg, where there are lecture halls, computer rooms and group rooms. Here you can also find student lunch room and reading room, as well as student counsellors and student office.
