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

TL;DR

This paper rigorously proves the formulae for the mean and covariance of Gaussian and other stochastic processes under unbounded linear operators, addressing technical gaps in existing literature.

Contribution

It provides a self-contained proof for the transformation of stochastic processes under closed, densely-defined operators, extending the theoretical foundation for Gaussian process transformations.

Findings

Rigorous proof of mean and covariance transformation formulas

Addresses technical gaps for unbounded operators

Utilizes Hille's theorem for Bochner integrals

Abstract

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.