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 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). © 1993 Springer-Verlag.