Question regarding the Wick tensor in white noise analysis

Question

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.

Answer 1

There is a lot of confusion around the concept of "Wick" product. Much of it is due to the following. As you mention, there is a general formula for the Wick product of a collection of random variables. Given a collection $\{X_i\}_{i \in I}$ and an $I$-valued multiindex $\alpha$, it says that $X^{\diamond \alpha}$ is the unique polynomial of degree $\alpha$ in $X$ (in the sense that each homogeneous term is of degree at most $\alpha_i$ in $X_i$ for all $i \in I$) such that $X^{\diamond 0} = 1$, $\partial_i X^{\diamond \alpha} = \alpha_i X^{\diamond \alpha - e_i}$, and $\mathbb{E} X^{\diamond \alpha} = 0$ for all $\alpha \neq 0$.

One can show that this can always be inverted in the sense that every polynomial $Y = \sum_\alpha Y_\alpha X^\alpha$ with $Y_\alpha \in \mathbb{R}$ can be written as a 'Wick' polynomial $\sum_\alpha Y_\alpha' X^{\diamond\alpha}$ of the same degree and vice-versa. This extends to formal power series.

So far, all this doesn't really look like a product, but it is then natural to define a product $\diamond$ on all Wick polynomials of the $X$'s by postulating that $X^{\diamond \alpha} \diamond X^{\diamond \beta} := X^{\diamond (\alpha+\beta)}$. Again, this product actually makes perfect sense not just between Wick polynomials, but between formal power series, at least provided one has suitable control on their growth. This is why you can define the Wick product between some Hida distributions that aren't actually random variables.

Now to the confusing bit. What if we add a new random variable $Y$ to the mix and would like to define for example $Y \diamond X$ where $X$ is one of the preexisting elements of our collection? On one hand, we could simply enlarge our collection to include $Y$ in which case one would set $Y \diamond X = XY - \mathbb{E} XY$ (assuming they are all centred). On the other hand, it may be that $Y$ is itself a polynomial function of the $X$'s or a limit of such functions. In this case, there's no need to extend our collection and we can simply use the previous definition. The two procedures will in general not give the same answer! Furthermore, in case $Y$ isn't a random variable at all but a formal power series in the $X$'s (e.g. a Hida distribution), only the second procedure would make sense while if $Y$ isn't measurable w.r.t. the $X$'s only the first procedure makes sense...