## The Annals of Probability

### The Radial Part of Brownian Motion on a Manifold: A Semimartingale Property

Wilfrid S. Kendall

#### Abstract

The usual Ito formula fails to apply for $r(X)$ when $r$ is a distance function and $X$ a Brownian motion on a general manifold, since $r$ fails to be differentiable on the cut-locus. It is shown that the discrepancy between the two sides of Ito's formula forms a monotonic random process (and hence is of locally bounded variation). In particular, $r(X)$ is a semimartingale.

#### Article information

Source
Ann. Probab., Volume 15, Number 4 (1987), 1491-1500.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176991988

Digital Object Identifier
doi:10.1214/aop/1176991988

Mathematical Reviews number (MathSciNet)
MR905343

Zentralblatt MATH identifier
0647.60086

JSTOR