White Noise Representation of Gaussian Random Fields

TL;DR

This paper establishes a representation theorem for Banach space valued Gaussian random variables as integrals against white noise, providing conditions for Gaussian fields to have such representations and extending existing integration theories.

Contribution

It introduces a new representation theorem for Gaussian random variables and fields, extending integration theory to more general index spaces.

Findings

Characterization of white noise representations for Gaussian fields

Necessary and sufficient conditions for existence of white noise representations

Extension of Gaussian process integration theory to compact measure spaces

Abstract

We obtain a representation theorem for Banach space valued Gaussian random variables as integrals against a white noise. As a corollary we obtain necessary and sufficient conditions for the existence of a white noise representation for a Gaussian random field indexed by a compact measure space. As an application we show how existing theory for integration with respect to Gaussian processes indexed by $[0,1]$ can be extended to Gaussian fields indexed by compact measure spaces.