Home | Current | Past volumes | About | Login | Notify | Contact | Search
 Probability Surveys > Vol. 13 (2016) open journal systems 

Fractional Gaussian fields: A survey

Asad Lodhia, Massachusetts Institute of Technology
Scott Sheffield, Massachusetts Institute of Technology
Xin Sun, Massachusetts Institute of Technology
Samuel S. Watson, Massachusetts Institute of Technology

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.

AMS 2000 subject classifications: 60G15, 60G60.

Creative Common LOGO

Full Text: PDF

Lodhia, Asad, Sheffield, Scott, Sun, Xin, Watson, Samuel S., Fractional Gaussian fields: A survey, Probability Surveys, 13, (2016), 1-56 (electronic). DOI: 10.1214/14-PS243.


[Adl10]     R. J. Adler. The geometry of random fields. Society for Industrial and Applied Mathematics, 2010. MR3396215

[AT07]     R. J. Adler and J. E. Taylor. Random fields and geometry, volume 115. Springer, 2007. MR2319516

[Bas98]     R. F. Bass. Diffusions and elliptic operators. Springer, 1998. MR1483890

[BGR61]     R. Blumenthal, R. Getoor, and D. Ray. On the distribution of first hits for the symmetric stable processes. Transactions of the American Mathematical Society, 99(3):540–554, 1961. MR0126885

[BGW83]     R. Bhattacharya, V. K. Gupta, and E. Waymire. The hurst effect under trends. Journal of Applied Probability, pages 649–662, 1983. MR0713513

[Bil99]     P. Billingsley. Convergence of Probability Measures, Wiley Series in Probability and Statistics. Wiley, New York, 1999. MR1700749

[Cap00]     P. Caputo. Harmonic Crystals: Statistical Mechanics and Large Deviations. PhD thesis, TU Berlin 2000, http://edocs.tu-berlin.de/diss/index.html, 2000.

[CD09]     J.-P. Chiles and P. Delfiner. Geostatistics: modeling spatial uncertainty, volume 497. John Wiley & Sons, 2009. MR2850475

[CDDS11]     A. Capella, J. Dávila, L. Dupaigne, and Y. Sire. Regularity of radial extremal solutions for some non-local semilinear equations. Communications in Partial Differential Equations, 36(8):1353–1384, 2011. MR2825595

[CG11]     S.-Y. A. Chang and M. d. M. González. Fractional Laplacian in conformal geometry. Advances in Mathematics, 226(2):1410–1432, 2011. MR2737789

[CI13]     S. Cohen and J. Istas. Fractional fields and applications, volume 73. Springer, 2013. MR3088856

[CS98]     Z.-Q. Chen and R. Song. Estimates on Green functions and Poisson kernels for symmetric stable processes. Mathematische Annalen, 312(3):465–501, 1998. MR1654824

[CS07]     L. Caffarelli and L. Silvestre. An extension problem related to the fractional Laplacian. Communications in Partial Differential Equations, 32(8):1245–1260, 2007. MR2354493

[CSS08]     L. A. Caffarelli, S. Salsa, and L. Silvestre. Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Inventiones Mathematicae, 171(2):425–461, 2008. MR2367025

[CT10]     X. Cabré and J. Tan. Positive solutions of nonlinear problems involving the square root of the Laplacian. Advances in Mathematics, 224(5):2052–2093, 2010. MR2646117

[DNPV]     E. Di Nezza, G. Palatucci, and E. Valdinoci. Hitchhiker’s guide to the fractional Sobolev spaces. arXiv preprint arxiv:1104.4345.

[Dob79]     R. Dobrushin. Gaussian and their subordinated self-similar random generalized fields. The Annals of Probability, 1–28, 1979. MR0515810

[Dod03]     S. Dodelson. Modern cosmology. Amsterdam (Netherlands): Academic Press, 2003.

[DRSV]     B. Duplantier, R. Rhodes, S. Sheffield, and V. Vargas. Log-correlated Gaussian field: an overview. In preparation.

[DS11]     B. Duplantier and S. Sheffield. Liouville quantum gravity and KPZ. Inventiones Mathematicae, 185(2):333–393, 2011. MR2819163

[Dub09]     J. Dubédat. SLE and the free field: partition functions and couplings. Journal of the American Mathematical Society, 22(4):995–1054, 2009. MR2525778

[Dud02]     R. M. Dudley. Real analysis and probability, volume 74. Cambridge University Press, 2002. MR1932358

[dW51]     H. de Wijs. Statistics of ore distribution. part i: frequency distribution of assay values. Journal of the Royal Netherlands Geological and Mining Society, 13:365–375, 1951.

[dW53]     H. de Wijs. Statistics of ore distribution. part ii: theory of binomial distribution applied to sampling and engineering problems. Journal of the Royal Netherlands Geological and Mining Society, 15:125–24, 1953.

[Dyn80]     E. Dynkin. Markov processes and random fields. Bulletin of the American Mathematical Society, 3(3):975–999, 1980. MR0585179

[FJ98]     G. Friedlander and M. Joshi. Introduction to the theory of distributions. Cambridge University Press, Cambridge, 1998. MR1721032

[Fol99]     G. B. Folland. Real analysis: modern techniques and their applications, volume 361. Wiley, New York, 1999. MR1681462

[Gan67]     R. Gangolli. Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters. Ann. Inst. H. Poincaré Sect. B (N.S.), 3:121–226, 1967. MR0215331

[GGS10]     F. Gazzola, H.-C. Grunau, and G. Sweers. Polyharmonic boundary value problems: positivity preserving and nonlinear higher order elliptic equations in bounded domains. Number 1991. Springer, 2010. MR2667016

[Hör03]     L. Hörmander. The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Reprint of the second (1990) edition. Springer, Berlin, 2003.

[Jan97]     S. Janson. Gaussian Hilbert spaces, volume 129. Cambridge University Press, 1997. MR1474726

[Ken01]     R. Kenyon. Dominos and the Gaussian free field. Annals of Probability, 1128–1137, 2001. MR1872739

[Kol40]     A. N. Kolmogorov. Wienersche spiralen und einige andere interessante kurven im hilbertschen raum. In CR (Dokl.) Acad. Sci. URSS, volume 26, pages 115–118, 1940. MR0003441

[Kri10]     G. Kristensson. Second order differential equations: special functions and their classification. Springer, 2010. MR2682403

[Kuo96]     H.-H. Kuo. White noise distribution theory. CRC Press, 1996. MR1387829

[Kur07]     N. Kurt. Entropic repulsion for a class of Gaussian interface models in high dimensions. Stochastic Processes and Their Applications, 117(1):23–34, 2007. MR2287101

[Kur09]     N. Kurt. Maximum and entropic repulsion for a Gaussian membrane model in the critical dimension. The Annals of Probability, 37(2):687–725, 2009. MR2510021

[Lax02]     P. D. Lax. Functional analysis. John Wiley und Sons, 2002. MR1892228

[LD72]     N. S. Landkof and A. P. Doohovskoy. Foundations of modern potential theory. Springer-Verlag, Berlin, 1972. MR0350027

[Lév40]     M. P. Lévy. Le mouvement Brownien plan. American Journal of Mathematics, 62(1):487–550, 1940. MR0002734

[Lév45]     P. Lévy. Sur le mouvement Brownien dépendant de plusieurs paramètres. CR Acad. Sci. Paris, 220(420):3–1, 1945.

[Man75]     B. B. Mandelbrot. On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars. Journal of Fluid Mechanics, 72(03):401–416, 1975.

[MC06]     P. McCullagh and D. Clifford. Evidence for conformal invariance of crop yields. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 462(2071):2119–2143, 2006.

[McC02]     P. McCullagh. What is a statistical model? Annals of Statistics, 1225–1267, 2002. MR1936320

[McK63]     H. McKean, Jr. Brownian motion with a several-dimensional time. Theory of Probability & Its Applications, 8(4):335–354, 1963. MR0157407

[MO69]     S. A. Molchanov and E. Ostrovskii. Symmetric stable processes as traces of degenerate diffusion processes. Theory of Probability & Its Applications, 14(1):128–131, 1969. MR0247668

[Mon15]     D. Mondal. Applying Dynkin’s isomorphism: an alternative approach to understand the Markov property of the de wijs process. Bernoulli, 2015. MR3352044

[MP10]     P. Mörters and Y. Peres. Brownian motion, volume 30. Cambridge University Press, 2010. MR2604525

[MS]     J. Miller and S. Sheffield. Imaginary geometry III: reversibility of SLEκ for κ (4,8). 2012. arXiv preprint arXiv:1201.1498.

[MS12a]     J. Miller and S. Sheffield. Imaginary geometry I: interacting SLEs. arXiv preprint arXiv:1201.1496, 2012.

[MS12b]     J. Miller and S. Sheffield. Imaginary geometry II: reversibility of SLEκ(ρ12) for κ (0,4). arXiv preprint arXiv:1201.1497, 2012.

[MS13]     J. Miller and S. Sheffield. Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees. arXiv preprint arXiv:1302.4738, 2013.

[Mun99]     J. Munkres. Topology, 2nd edition. Prentice Hall, 1999.

[MVN68]     B. B. Mandelbrot and J. W. Van Ness. Fractional Brownian motions, fractional noises and applications. SIAM Review, 10(4):422–437, 1968. MR0242239

[MZ13]     I. Melbourne and R. Zweimüller. Weak convergence to stable Lévy processes for nonuniformly hyperbolic dynamical systems. arXiv preprint arXiv:1309.6429, 2013.

[New80]     C. Newman. Self-similar random fields in mathematical physics. In Proceedings Measure Theory Conference. DeKalb, Illinois, 1980.

[Olv10]     F. W. Olver. NIST handbook of mathematical functions. Cambridge University Press, 2010. MR2723248

[OW89]     M. Ossiander and E. C. Waymire. Certain positive-definite kernels. Proceedings of the American Mathematical Society, 107(2):487–492, 1989. MR1011824

[RV06]     B. Rider and B. Virág. The noise in the circular law and the Gaussian free field. arXiv preprint arXiv:math/0606663, 2006. MR2361453

[RV13]     R. Rhodes and V. Vargas. Gaussian multiplicative chaos and applications: a review. arXiv preprint arXiv:1305.6221, 2013. MR3274356

[RY99]     D. Revuz and M. Yor. Continuous martingales and Brownian motion, volume 293. Springer Verlag, 1999. MR1725357

[Sak03]     H. Sakagawa. Entropic repulsion for a Gaussian lattice field with certain finite range interaction. Journal of Mathematical Physics, 44:2939, 2003. MR1982781

[Sak12]     H. Sakagawa. On the free energy of a Gaussian membrane model with external potentials. Journal of Statistical Physics, 147(1):18–34, 2012. MR2922757

[She07]     S. Sheffield. Gaussian free fields for mathematicians. Probability Theory and Related Fields, 139(3-4):521–541, 2007. MR2322706

[She10]     S. Sheffield. Conformal weldings of random surfaces: SLE and the quantum gravity zipper. arXiv preprint arXiv:1012.4797, 2010. MR2251117

[Sil07]     L. Silvestre. Regularity of the obstacle problem for a fractional power of the Laplace operator. Communications on Pure and Applied Mathematics, 60(1):67–112, 2007. MR2270163

[Sim79]     B. Simon. Functional integration and quantum physics, volume 86. Academic Press, 1979. MR0544188

[Sko56]     A. Skorokhod. Limit theorems for stochastic processes. Theory of Probability & Its Applications, 1(3):261–290, 1956. MR0084897

[Sko57]     A. Skorokhod. Limit theorems for stochastic processes with independent increments. Theory of Probability & Its Applications, 2(2):138–171, 1957. MR0094842

[SS10]     O. Schramm and S. Sheffield. A contour line of the continuum Gaussian free field. Probability Theory and Related Fields, 1–34, 2010. MR3101840

[Ste70]     E. M. Stein. Singular integrals and differentiability properties of functions, volume 2. Princeton University Press, 1970. MR0290095

[SW71]     E. M. Stein and G. L. Weiss. Introduction to Fourier analysis on Euclidean spaces (PMS-32), volume 1. Princeton University Press, 1971. MR0304972

[SW13]     X. Sun and W. Wu. Uniform spanning forests and the bi-Laplacian Gaussian field. arXiv preprint arXiv:1312.0059v1, 2013.

[Tao10]     T. Tao. An epsilon of room, I: Real analysis, volume 117 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2010. Pages from year three of a mathematical blog. MR2760403

[Tri83]     H. Triebel. Theory of function spaces, volume 78 of Monographs in Mathematics. Birkhäuser Verlag, Basel, 1983. MR0781540

[Won70]     E. Wong. Stochastic processes in information and dynamical systems. New York: McGraw-Hill, 1970.

[Xia13]     Y. Xiao. Recent developments on fractal properties of Gaussian random fields. In Further Developments in Fractals and Related Fields, pages 255–288. Springer, 2013. MR3184196

[Yag57]     A. M. Yaglom. Some classes of random fields in n-dimensional space, related to stationary random processes. Theory of Probability & Its Applications, 2(3):273–320, 1957. MR0094844

Home | Current | Past volumes | About | Login | Notify | Contact | Search

Probability Surveys. ISSN: 1549-5787