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The structure of a Brownian bubble
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  • Published: December 1993

The structure of a Brownian bubble

  • Robert C. Dalang1 &
  • John B. Walsh2 

Probability Theory and Related Fields volume 96, pages 475–501 (1993)Cite this article

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  • 21 Citations

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Summary

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.

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References

  1. Dalang, R.C., Walsh, J.B.: Geography of the level sets of the Brownian sheet. Probab. Theory Relat. Fields (to appear, 1993)

  2. Doob, J.L.: Classical potential theory and its probabilistic counterpart. Berlin Heidelberg New York: Springer 1984

    Google Scholar 

  3. Evans, S.N.: Potential theory for a family of several Markov processes. Ann. Inst. Henri Poincaré23, 499–530 (1987)

    Google Scholar 

  4. Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam: North Holland 1981

    Google Scholar 

  5. Imhof, J.P.: Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications. J. Appl. Probab.21, 500–510 (1984)

    Google Scholar 

  6. Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. Berlin Heidelberg New York: Springer 1988

    Google Scholar 

  7. Meyer, P.A.: Théorie élémentaire des processes à deux indices. In: Korezlioglu, H., Mazziotto, G., Szpirglas, J. (eds.) Processus aléatoires à deux indices. (Lect. Notes Math., vol. 863, pp. 1–39) Berlin Heidelberg New York: Springer 1981

    Google Scholar 

  8. Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Berlin Heidelberg New York: Springer 1991

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Tufts University, 02155, Medford, MA, USA

    Robert C. Dalang

  2. Department of Mathematics, University of British Columbia, V6T1Y4, Vancouver, British Columbia, Canada

    John B. Walsh

Authors
  1. Robert C. Dalang
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  2. John B. Walsh
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Additional information

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

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Cite this article

Dalang, R.C., Walsh, J.B. The structure of a Brownian bubble. Probab. Th. Rel. Fields 96, 475–501 (1993). https://doi.org/10.1007/BF01200206

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  • Received: 31 March 1992

  • Revised: 01 March 1993

  • Issue Date: December 1993

  • DOI: https://doi.org/10.1007/BF01200206

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Mathematics subject classifications (1991)

  • 60G60
  • 60G15
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