Geometrical characteristics of the polygonal line associated with a trajectory of the Wiener process

The talk is devoted to the geometrical properties of the trajectories of the Wiener process. The main interest in the subject is inspired by the mathematical theory of polymers. It is known that the long molecules of linear polymers can be modeled as trajectories of non-smooth random processes. As a basic model Wiener trajectories are considered. The main problem arising there is the geometry of such random curves. We propose to study it considering the polygonal line inscribed into the trajectory. We will discuss the asymptotic behavior of a number of self-intersections in the line for dimension 2.