  We establish sparsity and summability results for coefficient sequences of
Wiener-Hermite polynomial chaos expansions of countably-parametric solutions of
linear elliptic and parabolic divergence-form partial differential equations
with Gaussian random field inputs.
  The novel proof technique developed here is based on analytic continuation of
parametric solutions into the complex domain. It differs from previous works
that used bootstrap arguments and induction on the differentiation order of
solution derivatives with respect to the parameters. The present
holomorphy-based argument allows a unified, ``differentiation-free'' proof of
sparsity (expressed in terms of $\ell^p$-summability or weighted
$\ell^2$-summability) of sequences of Wiener-Hermite coefficients in polynomial
chaos expansions in various scales of function spaces. The analysis also
implies corresponding analyticity and sparsity results for posterior densities
in Bayesian inverse problems subject to Gaussian priors on uncertain inputs
from function spaces.
  Our results furthermore yield dimension-independent convergence rates of
various \emph{constructive} high-dimensional deterministic numerical
approximation schemes such as single-level and multi-level versions of
Hermite-Smolyak anisotropic sparse-grid interpolation and quadrature in both
forward and inverse computational uncertainty quantification.
