  In the study of partial differential equations (PDEs) with random initial
data and singular stochastic PDEs with random forcing, we typically decompose a
classically ill-defined solution map into two steps, where, in the first step,
we use stochastic analysis to construct various stochastic objects. The
simplest kind of such stochastic objects is the Wick powers of a basic
stochastic term (namely a random linear solution, a stochastic convolution, or
their sum). In the case of randomized initial data of a general function of
negative regularity for studying nonlinear wave equations (NLW), we show
necessity of imposing additional Fourier-Lebesgue regularity for constructing
Wick powers by exhibiting examples of functions slightly outside $L^2(\mathbb
T^d)$ such that the associated Wick powers do not exist. This shows that
probabilistic well-posedness theory for NLW with general randomized initial
data fails in negative Sobolev spaces (even with renormalization). Similar
examples also apply to stochastic NLW and stochastic nonlinear heat equations
with general white-in-time stochastic forcing, showing necessity of appropriate
Fourier-Lebesgue $\gamma$-radonifying regularity in the construction of the
Wick powers of the associated stochastic convolution.
