Suppose a function $G:\mathbb{R}^d\rightarrow\mathbb{R}$ is given.

What are some necessary or sufficient conditions on $G$ for there to exist a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a jointly-measurable function $V:\Omega\times\mathbb{R}^d\rightarrow \mathbb{R}$ that realizes a spatially homogenous Gaussian random field with $\mathbb{E}(V(\cdot,x)) = 0$ and $\mathbb{E}(V(\cdot,x)V(\cdot,y)) = G(x-y)$?