The radial part of Brownian motion. II: Its life and times on the cut locus (Q1326349)

This paper is a sequel to the second author's paper in Ann. Probab. 15, 1491-1500 (1987; Zbl 0647.60086), 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 at 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 of \(L\) 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 \textit{H. Wu} [Acta Math. 142, 57-78 (1979; Zbl 0403.53022)] and applied to radial parts of \(\Gamma\)-martingales by the second author [``The radial part of a \(\Gamma\)-martingale and a non-implosion theorem'' Ann. Probab. (1993; to appear)].