Wick Powers of Paley-Wiener Functionals: Expansion in Cylindrical Hermite Polynomials, the Covariance Formula, Continuity in the Vector, and Membership in the Chaos

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with the Cameron-Martin space $H_{c}$, the Cameron-Martin squares $|h|_{c}^{2}$ and the Paley-Wiener functionals $\ell_{h}$ of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §cameron-martin; the Cameron-Martin inner product $\langle h,g\rangle_{c}$ of The Cameron-Martin Inner Product of Two Cameron-Martin Vectors and the Wick powers ${:}\ell_{h}^{n}{:}$ of The Wick Powers of a Paley-Wiener Functional; the multi-indices $\alpha\in\mathcal{A}_{n}$ with their factorials and powers, and the cylindrical Hermite polynomials $H_{\alpha}$; and the Wiener chaoses $\mathcal{H}_{n}$. For $h\in H_{c}$, $h/c$ denotes the real sequence $(h_{k}/c_{k})_{k\in\mathbb{N}}$. Let $h,g\in H_{c}$ and $m,n\in\mathbb{N}_{0}$.

1. (Square-integrability)¶ ${:}\ell_{h}^{n}{:}$ is Borel and $\int_{X}\bigl({:}\ell_{h}^{n}{:}\bigr)^{2}\,d\gamma_{c}<\infty$, so that ${:}\ell_{h}^{n}{:}\in L^{2}(\gamma_{c})$.

2. (Expansion for finitely many coordinates)¶ Let $N\in\mathbb{N}$ with $h_{k}=0$ for every $k>N$, and let $\mathcal{A}_{n,N}$ be the finite set of the $\alpha\in\mathcal{A}_{n}$ with $\alpha_{k}=0$ for every $k>N$. Then for every $x\in X$,

$${:}\ell_{h}^{n}{:}(x)=\sum_{\alpha\in\mathcal{A}_{n,N}}\frac{n!}{\alpha!}\,(h/c)^{\alpha}\,H_{\alpha}(x).$$

3. (Covariance)¶ The product ${:}\ell_{h}^{m}{:}\;{:}\ell_{g}^{n}{:}$ is integrable with respect to $\gamma_{c}$, and

$$\int_{X}{:}\ell_{h}^{m}{:}\;{:}\ell_{g}^{n}{:}\,d\gamma_{c}=\begin{cases}n!\,\langle h,g\rangle_{c}^{\,n}&\text{if }m=n,\\0&\text{if }m\ne n.\end{cases}$$

4. (Continuity in the vector)¶ Let $(h^{(j)})_{j\in\mathbb{N}}$ be a sequence in $H_{c}$ such that $h^{(j)}-h\in H_{c}$ for every $j$ and $\bigl(|h^{(j)}-h|_{c}^{2}\bigr)_{j\in\mathbb{N}}$ converges to $0$. Then $\bigl(\lVert{:}\ell_{h^{(j)}}^{n}{:}-{:}\ell_{h}^{n}{:}\rVert_{2}\bigr)_{j\in\mathbb{N}}$ converges to $0$.

5. (Chaos)¶ ${:}\ell_{h}^{n}{:}\in\mathcal{H}_{n}$.