the sense that $(-\Delta)^s$ can be written as a composition of $(-\Delta)^{s-\lfloor s\rfloor}$ and $(-\Delta)^{\lfloor s\rfloor}$.

For the properties of the poly-Laplacian, we refer the reader to [@gazzola2010polyharmonic] and references therein. For more properties of $(-{\Delta})^s$ where $s\in (0,1)$, see [@silvestre2007regularity] and reference therein.

White Noise {#subsec:white_noise}
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On a finite dimensional Hilbert space $\mathcal{H}$ with inner product $(\cdot,\cdot)_{\mathcal{H}}$, one characterization of standard Gaussian $h$ on $\mathcal{H}$ is that $h$ is a standard Gaussian in $\mathcal{H}$ if and only if for all $v\in \mathcal{H}$, $(h,v)_{\mathcal{H}}$ is a centered Gaussian variable with variance $(v,v)_\mathcal{H}$. If $\mathcal{H}$ is infinite dimensional, then it is not possible to define a random element of $\mathcal{H}$ that satisfies this condition [@janson1997gaussian; @sheffield2007gaussian]. Nevertheless, we can still say that a random functional (which we will denote by $(h,\cdot)_{\mathcal{H}}$) is a standard Gaussian on $\mathcal{H}$ if for all $v\in \mathcal{H}$, $(h,v)_{\mathcal{H}}$ is a centered Gaussian variable with variance $(v,v)_{\mathcal{H}}$. Note that such a functional cannot be almost surely continuous with respect to $\|\cdot
\|_{\mathcal{H}}$.

White noise on ${\mathbb{R}}^d$ can be regarded as a standard Gaussian on $L^2({\mathbb{R}}^d)$. We will define $W$ to be a random generalized function such that $(W,f)$ is a centered Gaussian with variance $\|f\|^2_{L^2({\mathbb{R}}^d)}$ for 