thus, we fix the relationship and reference both ${H}$ and $s$ throughout the paper. We note that the fields $\{\operatorname{FGF}_s({\mathbb{R}}^d)\,:\, s \in {\mathbb{R}}\}$ may be coupled with the same white noise so that holds for all $s\in {\mathbb{R}}$ (Proposition \[prop:FGF\_coupling\]).

Examples
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The simplest example of a fractional Gaussian field is $\operatorname{FGF}_0({\mathbb{R}}^d)$, which is white noise. We denote by ${\mathcal{S}}({\mathbb{R}}^d)$ the space of Schwartz functions on ${\mathbb{R}}^d$, and we let ${\mathcal{S}}'({\mathbb{R}}^d)$ be its dual, the space of tempered distributions (see Section \[sec:preliminaries\] for details). If $h\in {\mathcal{S}}'({\mathbb{R}}^d)$ and $\phi\in {\mathcal{S}}({\mathbb{R}}^d)$, we use the notation $(h,\phi)$ for $h$ evaluated at $\phi$. White noise (surveyed in [@kuo1996white]) is a random element of ${\mathcal{S}}'({\mathbb{R}}^d)$ with the property that for $\phi_1,\phi_2\in {\mathcal{S}}({\mathbb{R}}^d)$, the random variables $(h,\phi_1)$ and $(h,\phi_2)$ are centered Gaussians with covariance $$\operatorname{Cov}[(h,\phi_1),(h,\phi_2)] = \int_{{\mathbb{R}}^d}\phi_1(x)\phi_2(x)\,dx.$$

Taking $d=1$ and $s=1$, we see that $(-\Delta)^{-s/2}$ is the antiderivative operator. It follows that $\operatorname{FGF}_{1}({\mathbb{R}})$ is the antiderivative of one dimensional white noise, which is a Brownian motion interpreted as a real-valued function modulo constant. If we fix the constant by setting the value at 0, we get ordinary Brownian motion.

If $s=1$ and $d\in {\mathbb{N}}$, then $\operatorname{FGF}_{1}({\mathbb{R}}^d)$ is a $d$-dimensional generalization of Brownian motion called the 