Say I have a probability space $(\Omega, \Sigma, P)$ and two Gaussian Processes over this space $X_1, X_2$ such that: \begin{align*} X_1: \mathbb{R} \times \Omega \rightarrow \mathbb{R} \\ X_2: \mathbb{R} \times \Omega \rightarrow \mathbb{R} \end{align*}
Under what conditions (if any) is the process $X'(\omega, x) = X_1(\omega, X_2(\omega, x))$ also a Gaussian Process? Does this require that $X_1,X_2$ are independent, or just jointly normal?