is clear that if an $\operatorname{FGF}_s({\mathbb{R}})$, for $s \in (1/2,3/2]$, is defined modulo additive constant in a translation invariant way, then the distributional derivatives $\operatorname{FGF}_{s-1}({\mathbb{R}})$, $\operatorname{FGF}_{s-2}({\mathbb{R}})$, etc. are defined without an additive constant. Thus the $\operatorname{FGF}_s({\mathbb{R}})$ is defined as a random tempered distribution without an additive constant when $s \leq 1/2$. Similarly, if the $\operatorname{FGF}_s({\mathbb{R}})$, for $s \in (1/2,3/2]$ is defined modulo additive constant (in a translation invariant way), then the indefinite integrals $\operatorname{FGF}_{s+1}({\mathbb{R}})$, $\operatorname{FGF}_{s+2}({\mathbb{R}})$, etc. are respectively defined modulo linear polynomials, quadratic polynomials, etc.

The following proposition, rephrased and proved as Theorem \[thm: restriction\] in Section \[sec:restriction\], is one reason that the one-dimensional case is significant.

If ${H}\geq 0$, then the restriction of the $d$-dimensional FGF with Hurst parameter ${H}$ (i.e., with $s = {H}+ \frac{d}{2}$) to any fixed $k$-dimensional subspace (with $1 \leq k < d$) is a $k$-dimensional FGF with Hurst parameter ${H}$ (up to multiplicative constant).

Interpretation as a long range GFF {#sec:long-range-FGF}
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The Gaussian free field $\operatorname{FGF}_1({\mathbb{R}}^d)$ can be approximated by the discrete Gaussian free field, which only has nearest neighbor interactions. This discrete Markov property gives rise to the domain Markov property of the Gaussian free field in the limit [@sheffield2007gaussian]. In Section 