Abstract
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 language | English (US) |
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Pages (from-to) | 353-368 |
Number of pages | 16 |
Journal | Probability Theory and Related Fields |
Volume | 96 |
Issue number | 3 |
DOIs | |
State | Published - Sep 1993 |
Externally published | Yes |
All Science Journal Classification (ASJC) codes
- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty
Keywords
- Methematics Subject Classification (1991): 58G32, 60H10, 60J45