Fractional Gaussian fields: A survey
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1407.5598.pdf
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3.26 MB
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254f85b94390bcde410401026ae7f0e8
Author(s)
Lodhia, Asad Iqbal
Sheffield, Scott Roger
Sun, Xin
Watson, Samuel Stewart
Date Issued
February 2016
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
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1214/14-PS243
