Let $(X_t)_{t \in \mathbb{R}^2}$ be a Gaussian random field with mean function $m: \mathbb{R}^2 \rightarrow \mathbb{R}$ and covariance function $\rho: \mathbb{R}^2 \rightarrow \mathbb{R}$ and define another random field $(Y_t)_{t \in \mathbb{R}^2}$by $Y_t = X_t ^2$.
Let $Z$ be a C0x process with driving random measure $f(B) = \int_{B} Y_t dt$, with $B$ ranging through the planar Borel sets.
Now consider that $X_t$ is weakly stationary, then the mean function $m$ is constant and $\rho(u,v) = \rho(v-u)$.
I want to find $E[ f(u)]$ and $E[ f(u) f(v)]$. How can I do this with the provided information? We have a random measure $f(B)$ and should use a random intensity function to compute the expectation. Moreover, how can I compute the expectation with the integral?