Sparsity for Infinite-Parametric Holomorphic Functions on Gaussian Spaces

Abstract. We investigate the sparsity of Wiener polynomial chaos expansions of holomorphic maps [Formula: see text] on Gaussian Hilbert spaces. Representing the Gaussian random field input as an affine-parametric expansion, the nonlinear map becomes a countably parametric, deterministic holomorphic map of the coordinate sequence [Formula: see text]. We establish weighted summability results for the Wiener–Hermite coefficient sequences of images of affine-parametric expansions of the log-Gaussian input under [Formula: see text]. These results give rise to [Formula: see text]-term approximation rate bounds for the full range of input summability exponents [Formula: see text]. We show that these approximation rate bounds apply to parameter-to-solution maps for elliptic diffusion PDEs with lognormal coefficients.