The integral of a Gaussian process on a unit sphere

Question

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!

Answer 1

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 $$ \bE(X^2)= \int_{S\times S} \bE(\bG f_u\bG f_v) dudv=\int_{S\times S} k(u,v) dudv. $$