## Probability Surveys

### Fractional Gaussian fields: A survey

#### Abstract

We discuss a family of random fields indexed by a parameter $s\in\mathbb{R}$ which we call the fractional Gaussian fields, given by $\mathrm{FGF}_{s}(\mathbb{R} ^{d})=(-\Delta)^{-s/2}W,$ where $W$ is a white noise on $\mathbb{R}^{d}$ and $(-\Delta)^{-s/2}$ is the fractional Laplacian. These fields can also be parameterized by their Hurst parameter $H=s-d/2$. In one dimension, examples of $\mathrm{FGF}_{s}$ processes include Brownian motion ($s=1$) and fractional Brownian motion ($1/2<s<3/2$). Examples in arbitrary dimension include white noise ($s=0$), the Gaussian free field ($s=1$), the bi-Laplacian Gaussian field ($s=2$), the log-correlated Gaussian field ($s=d/2$), Lévy’s Brownian motion ($s=d/2+1/2$), and multidimensional fractional Brownian motion ($d/2<s<d/2+1$). These fields have applications to statistical physics, early-universe cosmology, finance, quantum field theory, image processing, and other disciplines.

We present an overview of fractional Gaussian fields including covariance formulas, Gibbs properties, spherical coordinate decompositions, restrictions to linear subspaces, local set theorems, and other basic results. We also define a discrete fractional Gaussian field and explain how the $\mathrm{FGF}_{s}$ with $s\in(0,1)$ can be understood as a long range Gaussian free field in which the potential theory of Brownian motion is replaced by that of an isotropic $2s$-stable Lévy process.

#### Article information

Source
Probab. Surveys, Volume 13 (2016), 1-56.

Dates
First available in Project Euclid: 22 February 2016

https://projecteuclid.org/euclid.ps/1456149586

Digital Object Identifier
doi:10.1214/14-PS243

Mathematical Reviews number (MathSciNet)
MR3466837

Zentralblatt MATH identifier
1334.60055

Subjects
Primary: 60G15: Gaussian processes 60G60: Random fields

#### Citation

Lodhia, Asad; Sheffield, Scott; Sun, Xin; Watson, Samuel S. Fractional Gaussian fields: A survey. Probab. Surveys 13 (2016), 1--56. doi:10.1214/14-PS243. https://projecteuclid.org/euclid.ps/1456149586

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