Nonlinear Gaussian transformation in Hilbert spaces

Abstract

A nonlinear generalization of the Gaussian integral transformation on separable real Hilbert spaces is proposed and studied. Such transformations arise in the description of hierarchical models realized as sequences of probability measures on Hilbert spaces generated by nonlinear transformations. The nonlinear Gaussian transformation is defined as a continuous map acting between Fréchet spaces of holomorphic functions on complexifications of the Hilbert spaces. A number of properties of such spaces and of the nonlinear Gaussian transformation are described. In particular, a family of the fixed points of the latter is obtained and the stability of these points is investigated.