Gaussian random fields on non-separable Banach spaces

TL;DR

This paper investigates the existence and properties of Gaussian random fields on non-separable Banach spaces, including spaces of measures and H"older functions, establishing conditions for their regularity and direct definitions.

Contribution

It introduces new conditions for Gaussian random fields on non-separable Banach spaces, enabling direct construction on measure spaces and linking regularity to covariance kernels.

Findings

Gaussian white noise can be defined directly on measure spaces.

H"older regularity of samples is determined by the covariance kernel.

Connections to Kolmogorov-Chentsov theorem are established.

Abstract

We study Gaussian random fields on certain Banach spaces and investigate conditions for their existence. Our results apply inter alia to spaces of Radon measures and H\"older functions. In the former case, we are able to define Gaussian white noise on the space of measures directly, avoiding, e.g., an embedding into a negative-order Sobolev space. In the latter case, we demonstrate how H\"older regularity of the samples is controlled by that of the covariance kernel and, thus, show a connection to the Theorem of Kolmogorov-Chentsov.