Fractional Gaussian fields: a survey

Understand

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 ($s = d/2$), L\'evy's Brownian motion ($s = d/2 + 1/2$), and multidimensional fractional Brownian motion ($d/2 < s < d/2 + 1$).