Karhunen-Lo\`eve decomposition of Gaussian measures on Banach spaces

TL;DR

This paper extends the Karhunen-Loève decomposition, originally for Hilbert spaces, to Gaussian measures on Banach spaces, providing a constructive generalization of spectral theorems for covariance operators.

Contribution

It introduces a novel, constructive method to decompose Gaussian measures on Banach spaces, generalizing the spectral theorem beyond Hilbert spaces.

Findings

Provides a constructive decomposition for Gaussian measures on Banach spaces

Generalizes the spectral theorem for covariance operators

Matches Lévy's Brownian motion construction in the Wiener measure case

Abstract

The study of Gaussian measures on Banach spaces is of active interest both in pure and applied mathematics. In particular, the spectral theorem for self-adjoint compact operators on Hilbert spaces provides a canonical decomposition of Gaussian measures on Hilbert spaces, the so-called Karhunen-L{\`o} eve expansion. In this paper, we extend this result to Gaussian measures on Banach spaces in a very similar and constructive manner. In some sense, this can also be seen as a generalization of the spectral theorem for covariance operators associated to Gaussian measures on Banach spaces. In the special case of the standard Wiener measure, this decomposition matches with Paul L{\'e}vy's construction of Brownian motion.