Fractional Gaussian fields: A survey
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 ca...
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Institute of Mathematical Statistics
2018
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| Online Access: | http://hdl.handle.net/1721.1/115331 https://orcid.org/0000-0002-6677-5349 https://orcid.org/0000-0002-5951-4933 https://orcid.org/0000-0002-8579-1686 |
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| author | Lodhia, Asad Iqbal Sheffield, Scott Roger Sun, Xin Watson, Samuel Stewart |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Lodhia, Asad Iqbal Sheffield, Scott Roger Sun, Xin Watson, Samuel Stewart |
| author_sort | Lodhia, Asad Iqbal |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T16:49:23Z |
| format | Article |
| id | mit-1721.1/115331 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T16:49:23Z |
| publishDate | 2018 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | mit-1721.1/1153312022-10-03T08:34:33Z Fractional Gaussian fields: A survey Lodhia, Asad Iqbal Sheffield, Scott Roger Sun, Xin Watson, Samuel Stewart Massachusetts Institute of Technology. Department of Mathematics Lodhia, Asad Iqbal Sheffield, Scott Roger Sun, Xin Watson, Samuel Stewart 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. National Science Foundation (U.S.) (Grant DMS 1209044) National Science Foundation (U.S.). Graduate Research Fellowship Program (Award 1122374) 2018-05-11T17:37:12Z 2018-05-11T17:37:12Z 2016-02 2014-09 2018-05-01T16:44:24Z Article http://purl.org/eprint/type/JournalArticle 1549-5787 http://hdl.handle.net/1721.1/115331 Lodhia, Asad, et al. “Fractional Gaussian Fields: A Survey.” Probability Surveys, vol. 13, no. 0, 2016, pp. 1–56. https://orcid.org/0000-0002-6677-5349 https://orcid.org/0000-0002-5951-4933 https://orcid.org/0000-0002-8579-1686 http://dx.doi.org/10.1214/14-PS243 Probability Surveys Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Institute of Mathematical Statistics arXiv |
| spellingShingle | Lodhia, Asad Iqbal Sheffield, Scott Roger Sun, Xin Watson, Samuel Stewart Fractional Gaussian fields: A survey |
| title | Fractional Gaussian fields: A survey |
| title_full | Fractional Gaussian fields: A survey |
| title_fullStr | Fractional Gaussian fields: A survey |
| title_full_unstemmed | Fractional Gaussian fields: A survey |
| title_short | Fractional Gaussian fields: A survey |
| title_sort | fractional gaussian fields a survey |
| url | http://hdl.handle.net/1721.1/115331 https://orcid.org/0000-0002-6677-5349 https://orcid.org/0000-0002-5951-4933 https://orcid.org/0000-0002-8579-1686 |
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