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A recurrent theme in functional analysis is the interplay between the theoryof positive definite functions, and their reproducing kernels, on the one hand,and Gaussian stochastic processes, on the other. This central theme ismotivated by a host of applications, e.g., in mathematical physics, and instochastic differential equations, and their use in financial models. In thispaper, we show that, for three classes of cases in the correspondence, it ispossible to obtain explicit formulas which are amenable to computations of therespective Gaussian stochastic processes. For achieving this, we first developtwo functional analytic tools. They are: $(i)$ an identification of a universalsample space $\Omega$ where we may realize the particular Gaussian processes inthe correspondence; and (ii) a procedure for discretizing computations in$\Omega$. The three classes of processes we study are as follows: Processesassociated with: (a) arbitrarily given sigma finite regular measures on a fixedBorel measure space; (b) with Hilbert spaces of sigma-functions; and (c) withsystems of self-similar measures arising in the theory of iterated functionsystems. Even our results in (a) go beyond what has been obtained previously,in that earlier studies have focused on more narrow classes of measures,typically Borel measures on $\mathbb R^n$. In our last theorem (section 10),starting with a non-degenerate positive definite function $K$ on some fixed set$T$, we show that there is a choice of a universal sample space $\Omega$, whichcan be realized as a "boundary" of $(T, K)$. Its boundary-theoretic propertiesare analyzed, and we point out their relevance to the study of electricalnetworks on countable infinite graphs.

Spectral theory for Gaussian processes: Reproducing kernels, random  functions, boundaries, and $\mathbf L^2$-wavelet generators with fractional  scales