Probability Seminar by Balint Virag

Brownian beads
Balint Virag
Abstract: Two-dimensional stochastic models are of great interest to physicists and probabilists because they reflect the rich world of conformal symmetries. We consider the simplest of these models, planar Brownian motion. Two and three are the only dimensions where Brownian motion has cut-times, that is times at which past and future paths are disjoint. Brownian beads are the sections of the path in between the cut-times. We show that these beads are independent of each other (after a conformal transformation) and planar Brownian motion can be "stringed" out of them in the same way that Brownian motion on the line is constructed from excursions.