Van Ness discuss various empirical studies of real world processes (the price of wheat, water flowing through the Nile, etc.) that had been made by Hurst, who found different scaling exponents in different settings[^6].

The definition of fractional Brownian motion can be extended to describe a random function modulo additive constant on ${\mathbb{R}}^d$ when $d > 1$. Given ${H}\in(0,1)$ we define the FBM (also called the fractional Brownian field) on ${\mathbb{R}}^d$ as a mean-zero Gaussian process $(B^{{H}}_t)_{t\in {\mathbb{R}}}$ with covariance $$\operatorname{Cov}(B^{H}_tB^{H}_s) = \frac{1}{2}(|t|^{2{H}}+|s|^{2{H}}-|t-s|^{2{H}}),$$ where ${H}$ is the Hurst parameter of the field. We will prove in Section \[sec:FBM\] that the multidimensional fractional Brownian motion defined this way is equivalent to $\operatorname{FGF}_s({\mathbb{R}}^d)$, where $H=s-\frac{d}{2}\in(0,1)$.

In the case ${H}= 1/2$, this multidimensional process was introduced by Lévy in 1940 and is known as Lévy Brownian motion [@levy1940mouvement]. General processes including multidimensional fractional Brownian motion are discussed in Yaglom in 1957 and by Gangolli in 1967 [@yaglom1957some; @MR0215331]. (Gangolli gives general analytic arguments for positive definiteness of covariance kernels that apply in this case.) Fractional Brownian motion is studied in more detail in works of Mandelbrot, as referenced in [@mandelbrot1975geometry]. More detailed and modern discussions of fractional Brownian motion (including topics such as 