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The radial part of Brownian motion II. Its life and times on the cut locus
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  • Published: September 1993

The radial part of Brownian motion II. Its life and times on the cut locus

  • Michael Cranston1,
  • Wilfrid S. Kendall2 &
  • Peter March3 

Probability Theory and Related Fields volume 96, pages 353–368 (1993)Cite this article

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  • 7 Citations

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Summary

This paper is a sequel to Kendall (1987), which explained how the Itô formula for the radial part of Brownian motionX on a Riemannian manifold can be extended to hold for all time including those times a whichX visits the cut locus. This extension consists of the subtraction of a correction term, a continuous predictable non-decreasing processL which changes only whenX visits the cut locus. In this paper we derive a representation onL in terms of measures of local time ofX on the cut locus. In analytic terms we compute an expression for the singular part of the Laplacian of the Riemannian distance function. The work uses a relationship of the Riemannian distance function to convexity, first described by Wu (1979) and applied to radial parts of Γ-martingales in Kendall (1993).

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Author information

Authors and Affiliations

  1. Mathematics Department, University of Rochester, 14627, Rochester, NY, USA

    Michael Cranston

  2. Statistics Department, University of Warwick, CV4 7AL, Coventry, UK

    Wilfrid S. Kendall

  3. Mathematics Department, Ohio State University, 43210, Columbus, OH, USA

    Peter March

Authors
  1. Michael Cranston
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  2. Wilfrid S. Kendall
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  3. Peter March
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Additional information

The first author's research was supported by a visiting fellowship awarded by the UK Science and Engineering Council, by travel funds provided by a European Community SCIENCE initiative, by the Max-Planck-Institute of Bonn, and by a grant from NSA

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Cranston, M., Kendall, W.S. & March, P. The radial part of Brownian motion II. Its life and times on the cut locus. Probab. Th. Rel. Fields 96, 353–368 (1993). https://doi.org/10.1007/BF01292677

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  • Received: 21 August 1992

  • Revised: 04 February 1993

  • Issue Date: September 1993

  • DOI: https://doi.org/10.1007/BF01292677

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Methematics Subject Classification (1991)

  • 58G32
  • 60H10
  • 60J45
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