# The integral of a Gaussian process on a unit sphere

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!

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

• Thanks a lot! It seems like a fantastic solution! Does there some theories to guarantee this?
– 香结丁
Nov 25, 2020 at 2:20
• Use Bochmer integrals to justify this. In fact one needs a thepry of integration of Banach valued functions to give a meaning to $$\int_S \mathbb{G} f_u du.$$ Nov 25, 2020 at 11:08
• Thank you so much! I have never touched these fantastic things before. Could you recommend some good textbooks or papers regarding these things for me? Thanks in advance!
– 香结丁
Nov 26, 2020 at 0:52
• Try the Chapter V of the book Functional Analysis by K. Yosida Nov 26, 2020 at 14:56
• I appreciate your generous help!
– 香结丁
Nov 27, 2020 at 1:32