## Fractional Gaussian fields: A survey

##### Author(s)

Lodhia, Asad Iqbal; Sheffield, Scott Roger; Sun, Xin; Watson, Samuel Stewart
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We discuss a family of random fields indexed by a parameter s ∈ R which we call the fractional Gaussian fields, given by FGF[subscript s](R[superscript d]) = (-Δ)[superscript -s/2]W, where W is a white noise on R[superscript d] and (-Δ)[superscript -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 FGF[subscript 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 FGF[subscript s] with s ∈ (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.

##### Date issued

2016-02##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Journal

Probability Surveys

##### Publisher

Institute of Mathematical Statistics

##### Citation

Lodhia, Asad, et al. “Fractional Gaussian Fields: A Survey.” Probability Surveys, vol. 13, no. 0, 2016, pp. 1–56.

Version: Author's final manuscript

##### ISSN

1549-5787