Fractional Gaussian fields: A survey

Abstract

We discuss a family of random fields indexed by a parameter s ∈ R which we call the fractional Gaussian fields, given bywhere W is a white noise on R d and (−∆) −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 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 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.

Conformal weldings of random surfaces: SLE and the quantum gravity zipper

“…the survey [RV14]. Furthermore, the restriction of a higher dimensional LGF to a (two-dimensional) plane yields a GFF on the plane, so that the higher dimensional LGF can be interpreted as a coupling of planar GFFs, one for each planar subspace; it remains unclear whether the imaginary geometry or level set structures corresponding to the individual slices can be unified in a coherent way [LSSW14].…”

Scaling Limits in Divisible Sandpiles: A Fourier Multiplier Approach

“…is strictly negative, then the limit is a random distribution while for H ∈ (k, k + 1), k ∈ N ∪ {0}, the field is a (k − 1)-differentiable function (Lodhia et al (2014), with the caveat that the results presented therein are worked out for R d or domains with zero boundary conditions). In the case of H ≥ 0, a stronger result could be pursued, namely an invariance principle à-la Donsker (as for example in Cipriani et al (2018a, Theorem 2.1), Cipriani et al (2018b, Theorem 3)).…”

Sharp Convergence of Nonlinear Functionals of a Class of Gaussian Random Fields

“…, 1), Ψ is simply the standard fractional Gaussian field (−∆) − β 2 ξ. We refer to the survey [LSSW16] for more details on fractional Gaussian fields. Example 1.6.…”