Sample paths properties of Gaussian fields with equivalent spectral densities

TL;DR

This paper demonstrates that Gaussian fields with equivalent spectral densities share identical sample path properties in certain Banach spaces, highlighting the spectral density's role in path behavior.

Contribution

It establishes a theoretical link between spectral density equivalence and sample path properties of Gaussian fields in Banach spaces.

Findings

Gaussian fields with equivalent spectral densities have the same sample path properties.

Sample path properties are preserved in any separable Banach space embedded in continuous functions.

The result applies to Gaussian fields over compact subsets of Euclidean space.

Abstract

We prove that if $X$ and $Y$ are two Gaussian fields with equivalent spectral densities, they have the same sample paths properties in any separable Banach space continuously embedded in $\mathcal{C}^0(K)$ where $K$ is a compact set of $\mathbb{R}^d$.