Random Walks in Random Surroundings

Davit Karagulyan received his doctoral degree in mathematics from KTH Royal Institute of Technology in 2017. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Dmitry Dolgopyat at the Department of Mathematics, University of Maryland, College Park, USA.

The project addresses issues in the field of dynamic systems, the branch of mathematics that can largely be described as deterministic chaos theory. The theory examines the processes that develop in time, for example certain physical systems.

One of the questions that laid the foundation for the study of dynamic systems was the three-body problem in celestial mechanics. This concerned the determination of the long-term trajectories for the Moon, the Earth, and the Sun. At the end of the 19th century, Henri Poincaré proved that no general solutions existed and the trajectories did not have to be stable. Today, studies of dynamic systems often involve many more bodies moving under specific conditions.

In one part of his project, Davit Karagulyan will address a classic problem regarding random walks. This concerns the analysis of the statistical properties for the trajectory of a charged particle moving in a randomly varying electric field. Will, for example, the law of large numbers apply in this case? Roughly, the law states that the long-term results for the averages of random events should balance out in the end.

Another research project focuses on the statistical properties for the trajectory of a particle bouncing elastically off two parallel walls. One of the walls can move randomly up and down. One question is then whether such a model can lead to arbitrarily high particle velocities and, if so, how common trajectories with such unlimited velocities would be.