Advanced topics in probability

Department of Mathematical Sciences

The course can be part of the following programmes: 1) Mathematical Sciences, Master's Programme (N2MAT) and 2) Bachelor's Programme in Mathematics (N1MAT).

Main field of studies: Mathematical Statistics.

Specialization: A1F, Second cycle, has second-cycle course/s as entry requirements.

Knowledge corresponding to the courses MSG110 Probability theory and MMA110 Integration theory.

Preliminary 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.

Learning outcomes 2025
On successful completion of the course the student will be able to:

• Prove the existence of Brownian motion.

• State, understand, and prove some of the basic properties of Brownian motion.

• Derive at least one of the arc sine laws and the passage time distribution for Brownian motion.

• Understand, state and prove some of the basic properties of Hausdorff dimension and Minkowski dimension.

• Prove Brownian motion in 2 or more dimensions has Hausdorff dimension 2.

• Understand the relationship between Brownian motion and the classical Dirichlet problem.

• Prove recurrence and transience results for Brownian motion and some of their self-intersection properties.

Lectures

Language of instruction: English

Hand in tasks and an oral exam.

If a student who has twice received a failing grade for the same examination component wishes to change examiner ahead of the next examination session, such a request should be made to the department in writing and should be approved by the department unless there are special reasons to the contrary (Chapter 6 Section 22 of the Higher Education Ordinance).

If a student has received a recommendation from the University of Gothenburg for study support for students with disabilities, the examiner may, where it is compatible with the learning outcomes of the course and provided that no unreasonable resources are required, decide to allow the student to sit an adjusted exam or alternative form of assessment.

In the event that a course has ceased or undergone major changes, students are to be guaranteed at least three examination sessions (including the ordinary examination session) over a period of at least one year, but no more than two years after the course has ceased/been changed. The same applies to internships and professional placements (VFU), although this is restricted to just one additional examination session.

The grading scale comprises Pass (G) and Fail (U).

The course will be evaluated at the end of the course together with the students. This will be followed by an anonymous survey.

The results of and possible changes to the course will be shared with students who participated in the evaluation and students who are starting the course.