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 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）是空间统计与机器学习中广泛使用的工具。对于线性变换$T$（作用于$u$的样本路径）下另一高斯过程$u$的像$v$，其均值函数与协方差核的公式几乎已成为学界公认的常识。然而这些公式在应用时往往缺乏对技术细节的严格关注，尤其当$T$为微分算子等无界算子（这在现代应用中十分常见）时。本文针对闭的、稠密定义算子$T$作用于平方可积（不限于高斯）随机过程样本路径的情形，给出了所述公式的自包含证明。我们的证明技术基于Banach值随机变量Bochner积分的Hille定理。