Gaussian processes (GPs) are widely-used tools in spatial statistics and machine learning and the formulae for the mean function and covariance kernel of a GP $v$ that is the image of another GP $u$ under a linear transformation $T$ acting on the sample paths of $u$ are well known, almost to the point of being folklore. However, these formulae are often used without rigorous attention to technical details, particularly when $T$ is an unbounded operator such as a differential operator, which is common in several modern applications. This note provides a self-contained proof of the claimed formulae for the case of a closed, densely-defined operator $T$ acting on the sample paths of a square-integrable stochastic process. Our proof technique relies upon Hille's theorem for the Bochner integral of a Banach-valued random variable.

翻译：高斯过程（GP）是空间统计和机器学习中广泛使用的工具，对于受线性变换 $T$ 作用后，另一高斯过程 $u$ 的像 $v$ 的均值函数和协方差核的公式，已知程度近乎成为民间定理。然而，这些公式在使用时通常未严格关注技术细节，尤其当 $T$ 为微分算子等无界算子时（这在现代应用中常见）。本文针对闭稠定算子 $T$ 作用于平方可积随机过程样本路径的情形，提供了这些公式的自包含证明。我们的证明技术依赖于希尔定理（针对巴拿赫值随机变量的博赫纳积分）。