XVII: 21, 194-197, LNM 986 (1983)
PRICE, Gareth C.;
WILLIAMS, David
Rolling with `slipping': I (
Stochastic calculus,
Stochastic differential geometry)
If $Z$ and $\tilde Z$ are two Brownian motions on the unit sphere for the filtration of $Z$, there differentials $\partial Y=(\partial Z) \times Z$ (Stratonovich differentials and vector product) and $\partial\tilde Y$ (similarly defined) are related by $d\tilde Y = H dY$, where $H$ is a previsible, orthogonal transformation such that $HZ=\tilde Z$
Keywords: Brownian motion in a manifold,
Previsible representationNature: Original
Retrieve article from Numdam
XVII: 22, 198-204, LNM 986 (1983)
KARANDIKAR, Rajeeva L.
Girsanov type formula for a Lie group valued Brownian motion (
Brownian motion,
Stochastic differential geometry)
A formula for the change of measure of a Lie group valued Brownian motion is stated and proved. It needs a Borel correspondence between paths in the Lie algebra and paths in the group, that transforms all (continuous) semimartingales in the algebra into their stochastic exponential
Comment: For more on stochastic exponentials in Lie groups, see Hakim-Dowek-Lépingle
2023 and Arnaudon
2612Keywords: Changes of measure,
Brownian motion in a manifold,
Lie groupNature: Original Retrieve article from Numdam
XXIV: 30, 448-452, LNM 1426 (1990)
ÉMERY, Michel;
LÉANDRE, Rémi
Sur une formule de Bismut (
Markov processes,
Stochastic differential geometry)
This note explains why, in Bismut's work on the index theorem, the reference measure is not the Riemannian measure $r$ on the manifold, but $p_1(x,x) r(dx)$, where $p_t(x,y)$ is the density (with respect to $r$!) of the Brownian semi-group
Keywords: Brownian bridge,
Brownian motion in a manifold,
Transformations of Markov processesNature: Exposition,
Original additions Retrieve article from Numdam
XXXI: 05, 54-61, LNM 1655 (1997)
FANG, Shizan;
FRANCHI, Jacques
A differentiable isomorphism between Wiener space and path group (
Malliavin's calculus)
The Itô map $I$ is known to realize a measurable isomorphism between Wiener space $W$ and the group ${\cal P}$ of paths with values in a Riemannian manifold. Here, the pullback $I^{*}$ is shown to be a diffeomorphism (in the sense of Malliavin derivatives) between the exterior algebras $\Lambda (W)$ and $\Lambda ({\cal P})$. This allows to transfer the Weitzenböck-Shigekawa identity from $\Lambda (W)$ to $\Lambda ({\cal P})$, yielding for example the de~Rham-Hodge-Kodaira decomposition on ${\cal P}$
Keywords: Wiener space,
Path group,
Brownian motion in a manifold,
Differential formsNature: Original Retrieve article from Numdam
