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Geography of the level sets of the Brownian sheet
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  • Published: June 1993

Geography of the level sets of the Brownian sheet

  • Robert C. Dalang1 &
  • John B. Walsh2 

Probability Theory and Related Fields volume 96, pages 153–176 (1993)Cite this article

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  • 32 Citations

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Summary

We describe geometric properties of {W>α}, whereW is a standard real-valued Brownian sheet, in the neighborhood of the first hitP of the level set {W>α} along a straight line or smooth monotone curveL. In such a neighborhood we use a decomposition of the formW(s, t)=α−b(s)+B(t)+x(s, t), whereb(s) andB(t) are particular diffusion processes andx(s, t) is comparatively small, to show thatP is not on the boundary of any connected component of {W>α}. Rather, components of this set form clusters nearP. An integral test for thorn-shaped neighborhoods ofL with tip atP that do not meet {W>α} is given. We then analyse the position and size of clusters and individual connected components of {W>α} near such a thorn, giving upper bounds on their height, width and the space between clusters. This provides a local picture of the level set. Our calculations are based on estimates of the length of excursions ofB andb and an accounting of the error termx.

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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 NSF grant DMS-9103962, and, during the period of revision, by grant DAAL03-92-6-0323 from the Army Research Office

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Dalang, R.C., Walsh, J.B. Geography of the level sets of the Brownian sheet. Probab. Th. Rel. Fields 96, 153–176 (1993). https://doi.org/10.1007/BF01192131

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  • Received: 31 March 1992

  • Revised: 04 January 1993

  • Issue Date: June 1993

  • DOI: https://doi.org/10.1007/BF01192131

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Mathematics Subject Classifications (1991)

  • 60G60
  • 60G15
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