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The Mean Perimeter of Some Random Plane Convex Sets Generated by a Brownian Motion

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Abstract

If C 1 is the convex hull of the curve of a standard Brownian motion in the complex plane watched from 0 to 1, we consider the convex hulls of C 1 and several rotations of it and compute the mean of the length of their perimeter by elementary calculations. This can be seen geometrically as a study of the exit time by a Brownian motion from certain polytopes having the unit circle as an inscribed one.

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Authors and Affiliations

  1. CNRS, Laboratoire d’Informatique, Institut Gaspard Monge, Université Paris-Est, 5 Boulevard Descartes, Champs-sur-Marne, 77454, Marne-la-Vallée Cedex 2, France

    Philippe Biane

  2. Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 31062, Toulouse, France

    Gérard Letac

Authors
  1. Philippe Biane
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  2. Gérard Letac
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Correspondence to Gérard Letac.

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Biane, P., Letac, G. The Mean Perimeter of Some Random Plane Convex Sets Generated by a Brownian Motion. J Theor Probab 24, 330–341 (2011). https://doi.org/10.1007/s10959-009-0272-0

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  • DOI: https://doi.org/10.1007/s10959-009-0272-0

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