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(GP)是空间统计与机器学习中广泛使用的工具。关于GP $u$在线性变换$T$作用于其样本路径后的像$T u$的均值函数与协方差核的公式已广为人知，几乎成为常识。然而，这些公式常被使用而未严格关注技术细节，尤其当$T$为微分算子等无界算子时——这在现代应用中十分常见。本文对闭稠定算子$T$作用于平方可积(不必为高斯)随机过程样本路径的情形，给出所述公式的自包含证明。我们的证明方法依赖于希尔定理关于Banach值随机变量的Bochner积分。