The $\chi^2$ random field $U(t)$ with $n$ degree of freedom (dof) is defined as:
\begin{align} U(t) = \sum_{i=1}^n X_i(t)^2, t\in\mathbb{R}^N \end{align}
where $X_1(t),...,X_n(t)$ are i.i.d homogeneous, real-valued Gaussian random fields (GRF) with mean zero and unit variance. The finite dimensional distribution (fidi) of GRF is multivariate normal and the marginal distribution of $U(t)$ at each $t$ is $\chi^2$ with $n$ dof.
Can we say something about fidi of $U(t)$?