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

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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Correspondence to Nathan Ross.

Additional information

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

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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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Keywords

  • Brownian motion
  • Path decompositions
  • Convex minorant

Mathematics Subject Classification (2000)

  • 60J65
  • 60J05
  • 60E99