I have a question regarding the definition of Wick tensor in the framework of the white noise analysis.

To put some context to the question we start with the following Gel'fand triple

$$S(\mathbb R)\subset L^2(\mathbb R,dx)\subset S'(\mathbb R),$$

where $S$ and $S'$ are the Schwartz space of rapidly decreasing functions and its dual, the space of temperated distributions.

Let $(S',\mathcal B(S'),\mu)$ be the White noise probability space introduced by Hida.

Kuo introduces in this book the following notation where $:x^n:_{\sigma^2}$ stands for the $n$-th Hermite polynomial with parameter $\sigma^2$. After that the "Wick tensors" are introduced for elements in $x\in S'$.

My main doubt is: Does this construction has something to do with the Wick product as defined by Janson Svante in "Gaussian Hilbert spaces"?

As far as I know the Wick power $:f^n:$ can be defined for random variables $f$ with finite moments (Janson focuses on the case where the r.v. are Gaussian), but $x\in S'$ is not a random variable, actually $x$ is the "chance parameter"! (We can say that the action of $x$ on some test function is Gaussian though)

The formula he mentions above relating the Hermite polynomial and the Wick power is easily derived for the case of a centered Gaussian random variable, but again $x$ is not a Gaussian random variable!

Do you mind giving me some explanation for this? Thanks in advance.