On Kernels and Covariance Structures in Hilbert Space Gaussian Processes

TL;DR

This paper introduces a new framework for operator-valued kernels in Hilbert spaces, connecting operator theory and stochastic processes to develop Gaussian processes and analyze their covariance structures.

Contribution

It presents a comprehensive framework for operator-valued positive definite kernels, linking operator theory with stochastic processes and covariance analysis in Hilbert spaces.

Findings

Developed new Hilbert space-valued Gaussian processes

Analyzed covariance structures of these processes

Applied dilation constructions in operator theory

Abstract

Motivated by practical applications, I present a novel and comprehensive framework for operator-valued positive definite kernels. This framework is applied to both operator theory and stochastic processes. The first application focuses on various dilation constructions within operator theory, while the second pertains to broad classes of stochastic processes. In this context, the authors utilize the results derived from operator-valued kernels to develop new Hilbert space-valued Gaussian processes and to investigate the structures of their covariance configurations.