in which the potential theory of Brownian motion is replaced by that of an isotropic $2s$-stable Lévy process.
author:
- '<span style="font-variant:small-caps;">[Asad Lodhia[^1]]{}, [Scott Sheffield]{}, [Xin Sun]{}, [Samuel S. Watson[^2]]{}</span>'
title: 'Fractional Gaussian fields: a survey'
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Introduction {#sec:Introduction}
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![\[fig:indexgraph\] When ${H}< 0$ (grey shaded region) the FGF is defined as a random tempered distribution, not a function. When ${H}\in(0,1)$, the FGF is defined as a random continuous function modulo a global additive constant. Generally, for integers $k>0$ and ${H}\in(k,k+1)$, the FGF is a translation invariant random $k$-times-differentiable function defined modulo polynomials of degree $k$. For integer ${H}=k \geq 0$, the FGF is a random $(k-1)$-times-differentiable function (or distribution if $k=0$) defined modulo polynomials of degree $k$. ](./figures/indexgraph){width="3in"}

The $d$-dimensional **fractional Gaussian field** $h$ on ${\mathbb{R}}^d$ with index $s\in {\mathbb{R}}$ (abbreviated as $\operatorname{FGF}_s({\mathbb{R}}^d)$) is given by $$\label{eqn:fgflaplaciandef}
  h := (-\Delta)^{-s/2} W,$$ where $W$ is a real white noise on ${\mathbb{R}}^d$ and $ (-\Delta)^{-s/2}$ is the fractional Laplacian on ${\mathbb{R}}^d$. In Sections \[sec:preliminaries\] and \[sec:whole-space\], we will review classical and recent literature on the fractional Laplacian (see, e.g., [@landkof1972foundations; @silvestre2007regularity; @caffarelli2008regularity; @chang2011fractional]) and show how to assign rigorous meaning to .

Our goal 