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

Abstract

This paper is a sequel to Kendall (1987), which explained how the Ito 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).