Teaching

I am currently teaching a lecture course on the Brownian Web and Net. Please sign up for the course on JOGUStINe. If you have any questions, feel free to message me or visit me in 05-229

The Brownian Web and Net

Semester: Winter

Offered: 2024/2025

The Brownian motion is the universal scaling limit of a large class of one-dimensional random walks, and is of fundamental importance to probability theory and its applications. The Brownian web is comprised of uncountably many Brownian paths, interpreted as subsets of 1+1-dimensional continuous spacetime. Put simply, one Brownian path is started in every single point in spacetime and each pair of paths coalesces upon contact. Our first task will be to give a rigorous mathematical description of this object, which will be an interesting challenge in its own right. Subsequently, we will investigate the Brownian web more closely and discover a variety of intriguing properties and behaviours. Last not least, we will consider the convergence of collections of paths of discrete random walks to the Brownian web. 

The Brownian net is a variant of the Brownian web in which paths exhibit branching as well as coalescence. Here, our considerations will be mostly heuristic in nature.

This lecture presupposes a good basic knowledge of probability theory, as taught, for instance, in the lecture courses “Stochastics I & II”; the course “Stochastics II” may also be taken in parellel. A certain level of comfort with the more abstract notions of basic analysis, such as convergence in general metric spaces and (point-set) topology will be beneficial as well.