As the title said, how to calculate the expectation on the product of two random processes as follows:
$E\left[R(\mathbb{x})R'(\mathbb{x})\right]$, where $R(\mathbb{x})$ and $R'(\mathbb{x})$ represent two different Gaussian processes, and $\mathbb{x}$ is random variable vector.
The two different Gaussian processes are defined as:
$R(\mathbb{x})\sim \mathcal{N}(M(\mathbb{x}),C((\mathbb{x}))$
$R'(\mathbb{x})\sim \mathcal{N}(M'(\mathbb{x}),C'((\mathbb{x}))$
where $M(\mathbb{x})$ and $M'(\mathbb{x})$ are two different mean functions with respect to random variable vector $\mathbb{x}$; and $C(\mathbb{x})$ and $C'(\mathbb{x})$ are two different variance functions.