Maps that take Gaussian measures to Gaussian measures (Q1928882)

A structure theorem for Wiener maps which give rise to mappings that take centred Gaussian measures to centred Gaussian measures is proved. Namely, it is shown that a Borel measurable map \(\Phi:E\to F\) is a Wiener map if and only if the actions of \(\mathcal W\)-almost all \(y^*\in F^*\) on \(\Phi\) coincide with \(\mathcal I(A^Ty^*)\). Here, \(A^T\) is the adjoint of the bounded linear map \(A:H\to F\), \(H\) is a real separable Hilbert space continuously embedded in \(E\) as a dense subspace, the Paley-Wiener map \(\mathcal I\) is a linear isometry \(H\to L^2(\mathcal W,\mathbb R)\), and \(\mathcal W\) is a Borel probability measure on \(E\) making the triple \((H,E,\mathcal W)\) an abstract Wiener space. It is proved that such an \(A\) is unique and continuous from the weak* topology on \(H\) into the strong topology on \(F\). Consequently, necessary and sufficient properties of such an \(A\) are determined.