, Volume 96, Issue 4, pp 475-501

The structure of a Brownian bubble

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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)}, whereB is a standard Brownian motion andb 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 ofB(t)-b(s) in this component. These formulas are evaluated numerically and compared with the results from direct simulations ofB andb.

The research of this author was partially supported by grants DMS-9103962 from the National Science Foundation and DAAL03-92-6-0323 from the Army Research Office