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The change of variables formula on Wiener space. (English) Zbl 0889.60059

Azéma, J. (ed.) et al., Séminaire de probabilités XXXI. Berlin: Springer. Lect. Notes Math. 1655, 24-39 (1997).
The classical Jacobi formula of change of variables is given by \[ \int_{\mathbb{R}^n} \rho(x)g(Tx)|J(x)|\,dx= \int_{\mathbb{R}^n} g(x) \sum_{\theta\in T^{-1}(x)} \rho(\theta)\,dx, \] where \(T: \mathbb{R}^n\to \mathbb{R}^n\) is a \(C^1\)-map, \(J\) the Jacobian of \(T\) and \(\rho\), \(g\) are positive compactly supported bounded measurable functions. Here the Lebesgue measure can be replaced by a Gaussian probability. The mapping \(T: x\mapsto f(x)\) was generalized for the Lipschitz case under some conditions by H. Federer in his classical book on “Geometric measure theory” [Berlin: Springer Verlag (1969; Zbl 0176.00801)]. The authors state: “The transformations of measure induced by a not necessarily adapted perturbation of the identity is considered. Previous results are reviewed and recent results on absolutely continuity and related Radon-Nikodým densities are derived under conditions which are ‘as near as possible’ to [those] of Federer’s area theorem in the finite-dimensional case”.
For the entire collection see [Zbl 0864.00069].

MSC:

60H05 Stochastic integrals
60G30 Continuity and singularity of induced measures
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