## Decomposition of Gaussian processes, and factorization of positive definite kernels. (arXiv:1812.10850v1 [math.FA])

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.查看全文