  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.
