Spectral Theory for Gaussian Processes: Reproducing Kernels, Boundaries, and L²-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 in mathematical physics. In this article, we show that it is possible to obtain explicit formulas amenable to computations of the respective Gaussian stochastic processes. To achieve this, we first develop two functional analytic tools. These are the identification of a universal sample space Ω where we realize the particular Gaussian processes in the correspondence, a procedure for discretizing computations in Ω. Our processes are as follows: Processes associated with arbitrarily given sigma finite regular measures on a fixed Borel measure space, with Hilbert spaces of sigma-functions, and with systems of self-similar measures arising in the theory of iterated function systems. In our last theorem, 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 Ω 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.

Keywords:

• Bernoulli measures
• Boundary-representations
• Covariance
• Cuntz-relations
• Direct integral decompositions
• Gaussian processes
• Independence
• Iterated function systems
• Positive definite functions
• Reproducing kernels
• Reproducing kernel-Hilbert spaces
• Sigma-functions
• Wavelets

2000 Mathematics Subject Classification:

• 46F20
• 60G15
• 47B15
• 60H05
• 60H40

ACKNOWLEDGMENTS

Palle Jorgensen thanks the Department of Mathematics for hospitality. We wish also to thank our colleagues Dorin Dutkay (U.C. Florida), Rob Martin (Cape Town University), Paul Muhly (University of Iowa), Judy Packer (University of Colorado), Steen Pedersen (Wright St Univ), Myung-Sin Song (University of South Illinois), Feng Tian (Wright St University) for discussions.