The Radial Part of Brownian Motion on a Manifold: A Semimartingale Property
Kendall, Wilfrid S.
Ann. Probab., Tome 15 (1987) no. 4, p. 1491-1500 / Harvested from Project Euclid
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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.
Publié le : 1987-10-14
Classification:  Brownian motion,  Laplace-Beltrami operator,  cut-locus,  comparison theorem,  60J65,  58G32 
@article{1176991988,
     author = {Kendall, Wilfrid S.},
     title = {The Radial Part of Brownian Motion on a Manifold: A Semimartingale Property},
     journal = {Ann. Probab.},
     volume = {15},
     number = {4},
     year = {1987},
     pages = { 1491-1500},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176991988}
}

Kendall, Wilfrid S. The Radial Part of Brownian Motion on a Manifold: A Semimartingale Property. Ann. Probab., Tome 15 (1987) no. 4, pp.  1491-1500. http://gdmltest.u-ga.fr/item/1176991988/