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