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 $T u$ 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 many 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 (not necessarily Gaussian) 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（GPs）是空间统计和机器学习中广泛使用的工具，而作为另一个高斯过程 $u$ 在作用于其样本路径的线性变换 $T$ 下的像，GP $T u$ 的均值函数与协方差核的公式广为人知，几乎已成常识。然而，这些公式的使用往往缺乏对技术细节的严谨考量，尤其是当 $T$ 为无界算子（例如微分算子，这在许多现代应用中十分常见）时。本文针对闭的、稠定算子 $T$ 作用于平方可积（不一定是高斯）随机过程样本路径的情形，提供了所述公式的自包含证明。我们的证明技术依赖于希尔定理在巴拿赫值随机变量的博赫纳积分中的应用。