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

Abstract

A recurrent theme in functional analysis is the interplay between the theory of positive definite functions, and their reproducing kernels, on the one hand, and Gaussian stochastic processes, on the other. This central theme is motivated by a host of applications, e.g., in mathematical physics, and in stochastic differential equations, and their use in financial models. In this paper, we show that, for three classes of cases in the correspondence, it is possible to obtain explicit formulas which are amenable to computations of the respective Gaussian stochastic processes. For achieving this, we first develop two functional analytic tools. They are: $(i)$ an identification of a universal sample space $\Omega$ where we may realize the particular Gaussian processes in the correspondence; and (ii) a procedure for discretizing computations in $\Omega$. 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. 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 $\Omega$, which can be realized as a "boundary" of $(T, K)$. Its boundary-theoretic properties are analyzed, and we point out their relevance to the study of electrical networks on countable infinite graphs.