A universal envelope for Gaussian processes and their kernels

The central theme in our paper deals with mathematical issues involved in the answer to the following question: How can we generate stochastic processes from their correlation data? Since Gaussian processes are determined by moment information up to order two, we focus on the Gaussian case. Two functional analytic tools are used here, in more than one variant. They are: operator factorization; and direct integral decompositions in the form of Karhunen-Loève expansions. We define and study a new interplay between the theory of positive definition functions, and their reproducing kernels, on the one hand, and Gaussian stochastic processes, on the other. The three classes of processes we study are as follows: Processes associated with: (a) arbitrarily given sigma finite regular measures on a fixed Borel measure space; (b) with Hilbert spaces of sigma-functions; and (c) with systems of self-similar measures arising in the theory of iterated function systems. Starting with a non-degenerate positive definite function K on some fixed set S, we show that there is a choice of a universal sample space Ω, which can be realized as a “boundary” of (S,K). Its boundary-theoretic properties are analyzed, and we point out their relevance to the study of electrical networks on countable infinite graphs.