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

Michael Cranston, Wilfrid S. Kendall, Peter March

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


This paper is a sequel to Kendall (1987), which explained how the Itô formula for the radial part of Brownian motion X on a Riemannian manifold can be extended to hold for all time including those times a which X visits the cut locus. This extension consists of the subtraction of a correction term, a continuous predictable non-decreasing process L which changes only when X visits the cut locus. In this paper we derive a representation on L in terms of measures of local time of X 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).

Original languageEnglish (US)
Pages (from-to)353-368
Number of pages16
JournalProbability Theory and Related Fields
Issue number3
StatePublished - Sep 1993
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • Methematics Subject Classification (1991): 58G32, 60H10, 60J45


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