The structure of a Brownian bubble
We examine local geometric properties of level sets of the Brownian sheet, and in particular, we identify the asymptotic distribution of the area of sets which correspond to excursions of the sheet high above a given level in the neighborhood of a particular random point. It is equal to the area of certain individual connected components of the random set \(\{(s,t):B(t)>b(s)\}\), where \(B\) is a standard Brownian motion and \(b\) is (essentially) a Bessel process of dimension 3. This limit distribution is studied and, in particular, explicit formulas are given for the probability that a point belongs to a specific connected component, and for the expected area of a component given the height of the excursion of \(B(t)-b(s)\) in this component. These formulas are evaluated numerically and compared with the results from direct simulations of \(B\) and \(b\).
scientific article; zbMATH DE number 568993
scientific article