Decomposition of Gaussian processes, and factorization of positive definite kernels

We establish a duality for two factorization questions, one for general positive definite (p.d.) kernels \(K\), and the other for Gaussian processes, say \(V\). The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization for p.d. kernels is intuitively motivated by matrix factorizations, but in infinite dimensions, subtle measure theoretic issues must be addressed. Consider a given p.d. kernel \(K\), presented as a covariance kernel for a Gaussian process \(V\). We then give an explicit duality for these two seemingly different notions of factorization, for p.d. kernel \(K\), vs for Gaussian process \(V\). Our result is in the form of an explicit correspondence. It states that the analytic data which determine the variety of factorizations for \(K\) is the exact same as that which yield factorizations for \(V\). Examples and applications are included: point-processes, sampling schemes, constructive discretization, graph-Laplacians, and boundary-value problems