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XXXI: 05, 54-61, LNM 1655 (1997)

**FANG, Shizan**; **FRANCHI, Jacques**

A differentiable isomorphism between Wiener space and path group (Malliavin's calculus)

The ItĂ´ 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 WeitzenbĂ¶ck-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 forms

Nature: Original

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A differentiable isomorphism between Wiener space and path group (Malliavin's calculus)

The ItĂ´ 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 WeitzenbĂ¶ck-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 forms

Nature: Original

Retrieve article from Numdam