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Appendix to the preceding article: The natural filtration of the Brownian motion indexed by \(\mathbb{R}\) on a compact manifold. (Appendice à l’exposé précédent: La filtration naturelle du mouvement brownien indexé par \(\mathbb{R}\) dans une variété compacte.) (French) Zbl 0949.60089
Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXIII. Berlin: Springer. Lect. Notes Math. 1709, 304-314 (1999).
It is shown in a recent note of M. Émery and W. Schachermayer [ibid., 291-303 (1999)] that the natural filtration of a Brownian motion on the circle indexed by \({\mathbb{R}}\) is obtained from a usual Brownian filtration via a (deterministic) time-change. The purpose of this article is to prove the same result for Brownian motion on any compact Riemannian manifold of dimension \(d\geq 2\). The method relies on a generalization of results of M. Cranston [J. Funct. Anal. 99, No. 1, 110-124 (1991; Zbl 0770.58038)] and W. S. Kendall [Stochastics 19, 111-129 (1986; Zbl 0584.58045)] about the Brownian coupling property.
For the entire collection see [Zbl 0924.00016].
MSC:
60J65 Brownian motion
60G44 Martingales with continuous parameter
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