self-similar Gaussian random fields which were investigated and classified in [@dobrushin1979gaussian]. When $d=1$ and ${H}\in (0,1)$, the $\operatorname{FGF}_s({\mathbb{R}}^d)$ process is commonly known as [**fractional Brownian motion with Hurst parameter $H$**]{}, and is the subject of an extensive literature (see the survey [@cohenfracfield]). Brownian motion itself corresponds to $H=1/2$ and $s=1$.

The law of a Brownian motion or fractional Brownian motion $B_t$, indexed by $t \in {\mathbb{R}}$ and defined so that $B_0 = 0$, is not translation invariant. However, the law of Brownian motion is translation invariant if we consider Brownian motion as a random process defined only modulo a global additive constant. In other words, Brownian motion has stationary increments. Similarly, the indefinite integral of a Brownian motion can be interpreted, in a translation invariant way, as a random function defined modulo the space of linear functions. We generally interpret all of the $\operatorname{FGF}_s$ processes as translation invariant random distributions, but in some cases they are defined modulo a space of polynomials. More precisely, when ${H}< 0$, $\operatorname{FGF}_s({\mathbb{R}}^d)$ is a translation invariant random tempered distribution (that is, a generalized function) on ${\mathbb{R}}^d$. When ${H}> 0$, $\operatorname{FGF}_s({\mathbb{R}}^d)$ is a translation invariant random element of the space $C^{\lceil {H}\rceil - 1}({\mathbb{R}}^d)$ modulo 