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In the study of partial differential equations (PDEs) with random initialdata and singular stochastic PDEs with random forcing, we typically decompose aclassically ill-defined solution map into two steps, where, in the first step,we use stochastic analysis to construct various stochastic objects. Thesimplest kind of such stochastic objects is the Wick powers of a basicstochastic term (namely a random linear solution, a stochastic convolution, ortheir sum). In the case of randomized initial data of a general function ofnegative regularity for studying nonlinear wave equations (NLW), we shownecessity of imposing additional Fourier-Lebesgue regularity for constructingWick powers by exhibiting examples of functions slightly outside $L^2(\mathbbT^d)$ such that the associated Wick powers do not exist. This shows thatprobabilistic well-posedness theory for NLW with general randomized initialdata fails in negative Sobolev spaces (even with renormalization). Similarexamples also apply to stochastic NLW and stochastic nonlinear heat equationswith general white-in-time stochastic forcing, showing necessity of appropriateFourier-Lebesgue $\gamma$-radonifying regularity in the construction of theWick powers of the associated stochastic convolution.

A remark on randomization of a general function of negative regularity