Suppose there exist a zero-mean Gaussian process $\mathbb{G} f\_u$ indexed by $u \in \mathcal{S}^{p - 1}$ with known covariance $\mathrm{E} \big[ \mathbb{G} f\_u \mathbb{G} f\_v \big]$ when both $u$ and $v$ are known, where $\mathcal{S}^{p - 1}$ is the $p$-dimensional unit sphere. Now I want to know what exactly the integral
\begin{equation\*}
 \int\_{\mathcal{S}^{p - 1}} \, \mathbb{G} f\_u \, du
\end{equation\*}
is. This is a integral Gaussian process on the unit sphere. I try my best to find some articles about it, but I cannot find any useful information about it.


Does anyone can help me with how to handle this integral or know some literature about this integral? Thanks so much!



The integral of a Gaussian process on a unit sphere
Let $\newcommand{\bG}{\mathbb{G}}$ $\newcommand{\bE}{\mathbb{E}}$
$$
X=\int\_S \bG f\_u du,\;\;k(u,v)=\bE( \bG f\_u \bG f\_v).
$$
Then $X$ is a mean zero Gaussian random variable so it suffices to find its variance $\bE(X^2)$. Note that
$$
X^2=\int\_{S\times S} \bG f\_u\bG f\_v dudv
$$
so
$$
\bE(X^2)= \int\_{S\times S} \bE(\bG f\_u\bG f\_v) dudv=\int\_{S\times S} k(u,v) dudv.
$$


