I have an ensemble of Gaussian random fields $X(s)$ over coordinates $s$. The pdf of an individual realisation of the field taking a value $x(s)$ at coordinate $s$ is given by $$ f_x(x(s))=\frac{1}{(2\pi)^{N/2}|C_{xx}|^{1/2}}\exp{\left(-\frac{1}{2}x(s)^TC^{-1}_{xx}x(s)\right)}$$ where $C_{xx}$ is the covariance matrix over the coordinates.
I am interested in the ensemble of Fourier transforms of the Gaussian random fields, say $K(q)=\mathscr{F}[X(s)]$, where $q$ is the corresponding Fourier-space coordinate. In particular, I'd like a pdf that describes the realisations of the field in Fourier-space, $f_k(k(q))$, such that I can calculate the expected value of a some arbitrary function $\mathbb{E}[\phi(K(q))]$ over the ensemble. Is there a way to calculate such a function? And is there a way in the general case, for an arbitrary (non-Gaussian) random field?
There's the characteristic function, but as I understand it this is just an alternate representation of the pdf - it does not represent the Fourier transform of the field itself. Unfortunately, this is the only thing that shows up when I try to search this question.
My intuition is that $f_k(k(q))\propto \exp{[k(q)^T C_{xx} k(q)/2]}$ given that the diagonals needs to be the power spectrum of the original field but I'm not sure how to arrive there.
Thanks for the help!