Operator-valued Gaussian processes and their covariance kernels

Abstract

In this paper, we discuss a new framework for operator-valued Gaussian processes and their covariance kernels. Our emphasis is four-fold: (i) starting with a positive operator-valued measure (POVM) [Formula: see text], we present algorithms for constructing an associated centered, operator-valued, Gaussian process [Formula: see text] with [Formula: see text] as its covariance kernel; (ii) we present different classes of POVMs, and we examine the corresponding classes of operator-valued Gaussian processes [Formula: see text]; (iii) for the operator-valued Gaussian processes [Formula: see text] at hand, and the non-commutative framework, we present the corresponding Itô-integrals; and we (iv) outline features of the operator-valued setting which are different from the more familiar case of scalar-valued Gaussian processes.