The Wiener Chaoses of a Diagonal Gaussian Measure on a Hilbert Space

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with the set $\mathcal{A}$ of multi-indices, their orders $|\alpha|$ and the sets $\mathcal{A}_{m}$, and the cylindrical Hermite polynomials $H_{\alpha}$, whose classes lie in $L^{2}(\gamma_{c})$ by Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §cylindrical and are again written $H_{\alpha}$, by the convention of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §lebesgue. For a subset $S$ of $L^{2}(\gamma_{c})$, its linear span is the set consisting of the zero vector and of all finite sums $\sum_{i=1}^{r}t_{i}F_{i}$ with $r\in\mathbb{N}$, $t_{i}\in\mathbb{R}$ and $F_{i}\in S$; closures are taken in $L^{2}(\gamma_{c})$.

1. (Cylindrical Hermite span)¶ $\mathcal{H}_{\mathrm{fin}}$ is the linear span of $\{H_{\alpha}:\alpha\in\mathcal{A}\}$.

2. (Chaos of order $m$)¶ For $m\in\mathbb{N}_{0}$, the Wiener chaos of order $m$ of $\gamma_{c}$ is the closure $\mathcal{H}_{m}$ of the linear span of $\{H_{\alpha}:\alpha\in\mathcal{A}_{m}\}$.

3. (Chaos up to order $m$)¶ For $m\in\mathbb{N}_{0}$, $\mathcal{H}_{\le m}$ is the closure of the linear span of $\{H_{\alpha}:\alpha\in\mathcal{A},\ |\alpha|\le m\}$.