Üstünel, Ali Süleyman; Zakai, Moshe 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]. Reviewer: Malempati M. Rao (Riverside) Cited in 2 Documents MSC: 60H05 Stochastic integrals 60G30 Continuity and singularity of induced measures Keywords:absolutely continuous measures; change of variables; Radon-Nikodým densities; Federer’s area theorem Citations:Zbl 0634.49001; Zbl 0874.49001; Zbl 0176.00801 PDFBibTeX XMLCite \textit{A. S. Üstünel} and \textit{M. Zakai}, Lect. Notes Math. 1655, 24--39 (1997; Zbl 0889.60059) Full Text: Numdam EuDML