any line has the law of a Brownian motion (modulo additive constant). In dimension $5$, the Laplacian of Lévy Brownian motion is the Gaussian free field.

We also hope that this text will be a useful reference for experts in the study of Gaussian fields; to this end, we provide a robust account of the regularity of FGF fields, the long and short range correlation formulae, conditional expectations given field values outside of fixed domains, the Fourier transforms and spherical coordinate decompositions of the FGF, and various bounded-domain definitions of the FGF.

The family of fractional Gaussian fields includes several well-known Gaussian fields such as Brownian motion ($d=1$ and $s=1$), white noise ($s=0$), the Gaussian free field ($s=1$), and the log-correlated Gaussian field ($s =
\frac{d}{2}$).

Given $s\in {\mathbb{R}}$ and $d \geq 1$, the **Hurst parameter** $H$ is defined by $$\label{eqn:hdef}
  {H}\colonequals s-\frac{d}{2}.$$ The Hurst parameter describes a scaling relation satisfied by $h\sim
\operatorname{FGF}_s({\mathbb{R}}^d)$: for $a>0$, the field $x\mapsto h(a x)$ has the same law as $x\mapsto a^{H}h(x)$.[^4] Fields satisfying such a relation are said to be **self-similar**, and they arise naturally in the study of statistical physics models [@newman1980self]. The FGFs belong to a more general class of translation-invariant 