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A differentiable isomorphism between Wiener space and path group. (English) Zbl 0889.58084

Azéma, J. (ed.) et al., Séminaire de probabilités XXXI. Berlin: Springer. Lect. Notes Math. 1655, 54-61 (1997).
For a compact Lie group \(G\) endowed with left-invariant Cartan’s connection, the authors consider the path space \(\mathcal P\) and its Wiener measure \(\mathbb{P}\).
The main aim of the paper is to extend to \(({\mathcal P},\mathbb{P})\) the Shigekawa’s construction of the de Rham-Kodaira theorem on abstract Wiener spaces [I. Shigekawa, J. Math. Kyoto Univ. 26, No. 2, 191-202 (1986; Zbl 0611.58006)]. To do this, the authors involve the pullback \(I^*\) by the Ito map \(I\), a measurable isomorphism between the Wiener space \((W,\mu)\) and \(({\mathcal P},\mathbb{P})\), and having noticed the flatness of \(\mathcal P\), they show that \(I^*\) supplies a diffeomorphism between the differential structures of the exterior algebras \(\Lambda(W)\) and \(\Lambda(\mathcal P)\).
For the entire collection see [Zbl 0864.00069].

MSC:

58J65 Diffusion processes and stochastic analysis on manifolds
60G99 Stochastic processes

Citations:

Zbl 0611.58006
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