Images of Gaussian and other stochastic processes under closed, densely-defined, unbounded linear operators

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 Tu 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.