Probability Theory and Related Fields

, Volume 96, Issue 2, pp 153–176 | Cite as

Geography of the level sets of the Brownian sheet

  • Robert C. Dalang
  • John B. Walsh


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.

Mathematics Subject Classifications (1991)

60G60 60G15 


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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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