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（GPs）是空间统计学与机器学习中广泛使用的工具。当一个高斯过程 $u$ 经过作用于其样本路径的线性变换 $T$ 后得到新过程 $v$，关于 $v$ 的均值函数与协方差核的公式早已广为人知，几乎成为常识性结论。然而，这些公式的使用常缺乏对技术细节的严谨考量，尤其在 $T$ 为无界算子（如现代应用中常见的微分算子）时。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，给出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（针对Banach值随机变量的Bochner积分）。