Nonlinear transformations of the canonical Gauss measure on Hilbert space and absolute continuity
Let \((H, B, \mu)\) be an abstract Wiener space. The present paper investigates the absolute continuity of the cylindrical measure induced by the nonlinear transformation \(T(h)= h+ F(h)\), \(h\in H\), of the canonical Gauss measure on \(H\). The result is stated in terms of absolute continuity of \(\mu\circ \widetilde {T}^{-1}\) w.r.t. \(\mu\), where \(\widetilde {T}\omega= \omega+ \widetilde{F} (\omega)\), \(\omega\in B\), denotes the lifting of \(T\) onto \(B\). If, under a weak Gateaux- differentiability assumption on \(F\), for some increasing sequence of finite-dimensional orthogonal projectors \(P_ n\) acting on \(H\), \(P_ n\uparrow I\), the transformations \(T_ n(h)= h+ P_ n F(P_ n k)\), \(h\in H\), are bijective, all the corresponding Carleman-Fredholm determinants are everywhere different from zero and the densities of \(\widetilde{T}_ n\) w.r.t. \(\mu\) form a uniformly integrable family, then \(\mu_ 0 \widetilde{T}^{-1} \ll\mu\) and the corresponding density is given by Kusuoka's formula. In the sequel the authors study the stochastic integral occuring in this formula in some detail and provide sufficient conditions for the uniform integrability. The requirement of the existence of a sequence \(T_ n\) of 1-1 transformations approximating \(T\) corresponds to the assumption of \textit{D. Nualart} and \textit{M. Zakai} [Probab. Theory Relat. Fields 73, 255-280 (1986; Zbl 0601.60053)] that \(\widetilde {T}\) can be approximated by a sequence of smooth finite-dimensional 1-1 transformations \(\widetilde {T}_ n\). But the problem which continues to remain open is the presentation of general assumptions which guarantee the existence of such an approximating sequence.
nonlinear transformation of Wiener measure
scientific article; zbMATH DE number 627452
scientific article