THE RADIAL PART OF BROWNIAN-MOTION .2. ITS LIFE AND TIMES ON THE CUT LOCUS

This paper is a sequel to Kendall (1987), which explained how the Ito formula for the radial part of Brownian motion X on a Riemannian manif old can be extended to hold for all time including those times at whic h X visits the cut locus. This extension consists of the subtraction o f a correction term, a continuous predictable non-decreasing process L which changes only when X visits the cut locus. In this paper we deri ve a representation of L in terms of measures of local time of X on th e cut locus. In analytic terms we compute an expression for the singul ar 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 GAMMA-mar tingales in Kendall (1993).