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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 Gaussiandoi:10.7494/opmath.2019.39.4.497 fatcat:dsz72yqghbbzth4ozx7gmuoyn4