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The greatest convex minorant of Brownian motion, meander, and bridge
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  • Published: 23 August 2011

The greatest convex minorant of Brownian motion, meander, and bridge

  • Jim Pitman1 &
  • Nathan Ross1 

Probability Theory and Related Fields volume 153, pages 771–807 (2012)Cite this article

Abstract

This article contains both a point process and a sequential description of the greatest convex minorant of Brownian motion on a finite interval. We use these descriptions to provide new analysis of various features of the convex minorant such as the set of times where the Brownian motion meets its minorant. The equivalence of these descriptions is non-trivial, which leads to many interesting identities between quantities derived from our analysis. The sequential description can be viewed as a Markov chain for which we derive some fundamental properties.

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Authors and Affiliations

  1. University of California, Berkeley, 367 Evans Hall #3860, Berkeley, CA, 94720-3860, USA

    Jim Pitman & Nathan Ross

Authors
  1. Jim Pitman
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  2. Nathan Ross
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Corresponding author

Correspondence to Nathan Ross.

Additional information

J. Pitman’s research supported in part by N.S.F. Grant DMS-0806118.

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Cite this article

Pitman, J., Ross, N. The greatest convex minorant of Brownian motion, meander, and bridge. Probab. Theory Relat. Fields 153, 771–807 (2012). https://doi.org/10.1007/s00440-011-0385-0

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  • Received: 13 November 2010

  • Revised: 24 June 2011

  • Published: 23 August 2011

  • Issue Date: August 2012

  • DOI: https://doi.org/10.1007/s00440-011-0385-0

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Keywords

  • Brownian motion
  • Path decompositions
  • Convex minorant

Mathematics Subject Classification (2000)

  • 60J65
  • 60J05
  • 60E99
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