Probability Theory and Related Fields

, Volume 96, Issue 4, pp 475–501

The structure of a Brownian bubble

  • Robert C. Dalang
  • John B. Walsh

DOI: 10.1007/BF01200206

Cite this article as:
Dalang, R.C. & Walsh, J.B. Probab. Th. Rel. Fields (1993) 96: 475. doi:10.1007/BF01200206


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.

Mathematics subject classifications (1991)

60G60 60G15 

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Robert C. Dalang
    • 1
  • John B. Walsh
    • 2
  1. 1.Department of MathematicsTufts UniversityMedfordUSA
  2. 2.Department of MathematicsUniversity of British ColumbiaVancouverCanada

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