Intégration géométrique sur l’espace de Wiener. (Geometric integration on the Wiener space). (French) Zbl 0656.60046

Airault, H.; Malliavin, P.

Bull. Sci. Math., II. Sér. 112, No. 1, 3-52 (1988).

For a suitably smooth Wiener functional \(\Phi\) it is shown that the set \(V_{\xi}=\{w: \Phi (w)=\xi \}\) can be viewed as a submanifold in Wiener space, of which the finite codimension is the dimension of the range of \(\Phi\). The set \(V_{\xi}\) is defined only modulo slim sets, since the same is true for \(\Phi\). The set \(V_{\xi}\) arises as the support of the measure obtained by conditioning on \(\Phi =\xi\), and the measure can be obtained using Watanabe’s distributional version of Malliavin calculus. A co-area formula and a Stokes formula are obtained.

Reviewer: W.S.Kendall

MSC:

60G07	General theory of stochastic processes
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
58J65	Diffusion processes and stochastic analysis on manifolds

Keywords:

slim sets; Watanabe’s distributional version of Malliavin calculus; co- area formula; Stokes formula