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A critical function for the planar Brownian convex hull

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Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 1526)

Abstract

We prove that if the origin is translated so that the real axis is tangential to the (random) convex hull of a planar Brownian motion, touching at the origin, then for each positive \(\frac{{(\frac{\pi }{2} + \varepsilon )|x|\log ^3 (1/|x|)}}{{\log (1/|x|)}}\) is an upper function for the hull but \(\frac{{(\frac{\pi }{2} - \varepsilon )|x|\log ^3 (1/|x|)}}{{\log (1/|x|)}}\) is not.

AMS 1980 subject classification

  • 60G17
  • 60J65
  • Keywords and phrases
  • Brownian Motion
  • h-Process

Research partially supported by NSF Grant DMS-86-01800.

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References

  • Burdzy, K. and San Martin, J. (1989) Curvature of the convex hull of planar Brownian motion near its minimum point. Stochastic Processes and their Applications, 33, 89–103.

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© 1992 Springer-Verlag

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Mountford, T.S. (1992). A critical function for the planar Brownian convex hull. In: Azéma, J., Yor, M., Meyer, P.A. (eds) Séminaire de Probabilités XXVI. Lecture Notes in Mathematics, vol 1526. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084314

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  • DOI: https://doi.org/10.1007/BFb0084314

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-56021-0

  • Online ISBN: 978-3-540-47342-8

  • eBook Packages: Springer Book Archive