On Selected Geometric Properties of Brownian Motion Paths

Title: On Selected Geometric Properties of Brownian Motion Paths Author: Mgr. Ondřej Honzl E-mail Address: honzl@karlin.mff.cuni.cz Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Jan Rataj, CSc. E-mail Address: rataj@karlin.mff.cuni.cz Department: Mathematical Institute, Charles University Abstract: Our thesis is focused on certain geometric properties of Brownian motion paths. Firstly, it deals with cone points of Brownian motion in the plane and we show some connections between cone points and critical points of Brownian motion. The motivation of the study of critical points is provided by a pleasant behavior of the distance function outside of the set of these points. We prove the theorem on a non-existence of π+ cone points on fixed line. This statement leads us to the conjecture that there are only countably many critical points of the Brownian motion path in the plane. Next, the thesis discusses an asymptotic behavior of the surface area of r-neigh- bourhood of Brownian motion, which is called Wiener sausage. Using the proper- ties of a Kneser function, we prove the claim about the relation of the Minkowski content and S-content. As the consequence, we obtain a limit behavior of the surface area of the Wiener sausage almost surely in dimension d ≥ 3. Finally,..