Hilbert space valued Gaussian processes, their kernels, factorizations, and covariance structure

TL;DR

This paper introduces a new framework for operator-valued positive definite kernels, with applications to operator theory and the construction and analysis of Hilbert space-valued Gaussian processes and their covariance structures.

Contribution

It presents a novel approach to operator-valued kernels and demonstrates their use in creating and understanding Hilbert space-valued Gaussian processes.

Findings

Developed a general framework for operator-valued positive definite kernels

Constructed new Hilbert space-valued Gaussian processes

Analyzed covariance structures of these processes

Abstract

Motivated by applications, we introduce a general and new framework for operator valued positive definite kernels. We further give applications both to operator theory and to stochastic processes. The first one yields several dilation constructions in operator theory, and the second to general classes of stochastic processes. For the latter, we apply our operator valued kernel-results in order to build new Hilbert space-valued Gaussian processes, and to analyze their structures of covariance configurations.