The Annals of Probability

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

Wilfrid S. Kendall

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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

Permanent link to this document
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
links.jstor.org

Subjects
Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 58G32

Keywords
Brownian motion Laplace-Beltrami operator cut-locus comparison theorem

Citation

Kendall, Wilfrid S. The Radial Part of Brownian Motion on a Manifold: A Semimartingale Property. Ann. Probab. 15 (1987), no. 4, 1491--1500. doi:10.1214/aop/1176991988. https://projecteuclid.org/euclid.aop/1176991988


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