I am right now reading about the fractional Gaussian field and there came up the following question: What is the covariance Kernel of the fractional Gaussian field on $\mathbb{R}^d$ defined as a random element of the topological dual space of the Schwartz space. I couldnt find the solution in the internet. With covariance Kernel I mean a function
$C:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}$ such that for the fracional gaussian field h and for all $\phi,\psi\in\mathcal{S}(\mathbb{R}^d)\subseteq\mathcal{S}'(\mathbb{R}^d)$ we have
$\mathbb{E}[h(\phi)h(\psi)]=\displaystyle\int_{\mathbb{R}^d}\displaystyle\int_{\mathbb{R}^d}\phi(x)C(x,y)\psi(y)dxdy$.
Any help or source would be highly appreciated.
A classification in the rotation and translation invariant case is given in the introductory part of the thesis by Ajay Chandra "Construction and Analysis of a Hierarchical Massless Quantum Field Theory". See in particular Prop 2.9.
It is not correct to think of the covariance kernel $C$ as a function $\mathbb{R}^d\times \mathbb{R}^d\rightarrow\mathbb{R}^d$. It is a temperate distribution, i.e., it belongs to $\mathcal{S}'(\mathbb{R}^{2d})$. It has singular support on the diagonal $x=y$, so outside that diagonal it indeed corresponds to a function of the form constant times $\frac{1}{|x-y|^{2\kappa}}$. However, if $\kappa>\frac{d}{2}$ there is a complicated finite part to be taken to build a distribution from the function.