# A Karhunen-Lo\`{e}ve Theorem for Random Flows in Hilbert spaces

## TL;DR

This paper generalizes Mercer's theorem to infinite-dimensional Hilbert spaces and establishes a Karhunen-Loève expansion for mean-square continuous Hilbertian flows, enabling series representations with uncorrelated coefficients.

## Contribution

It introduces a novel generalization of Mercer's theorem and derives a Karhunen-Loève theorem for functional data in Hilbert spaces, extending classical results to infinite dimensions.

## Key findings

- Series expansion with uncorrelated coefficients for Hilbertian flows
- Uniform convergence of the expansion over time
- Extension of Mercer's theorem to operator-valued kernels in Hilbert spaces

## Abstract

We develop a generalisation of Mercer's theorem to operator-valued kernels in infinite dimensional Hilbert spaces. We then apply our result to deduce a Karhunen-Lo\`eve theorem, valid for mean-square continuous Hilbertian functional data, i.e. flows in Hilbert spaces. That is, we prove a series expansion with uncorrelated coefficients for square-integrable random flows in a Hilbert space, that holds uniformly over time.