Logo image
Stochastic Processes Induced by Singular Operators
Journal article   Peer reviewed

Stochastic Processes Induced by Singular Operators

Daniel Alpay and Palle E. T Jorgensen
Numerical Functional Analysis and Optimization: Operator Algebras and Representation Theory: Frames, Wavelets, and Fractals, Vol.33(7-9), pp.708-735
07/01/2012
DOI: 10.1080/01630563.2012.682132
url
https://digitalcommons.chapman.edu/cgi/viewcontent.cgi?article=1443&context=scs_articlesView
Open Access

Abstract

In this article, we study a general family of multivariable Gaussian stochastic processes. Each process is prescribed by a fixed Borel measure σ on ℝ n . The case when σ is assumed absolutely continuous with respect to Lebesgue measure was studied earlier in the literature, when n = 1. Our focus here is on showing how different equivalence classes (defined from relative absolute continuity for pairs of measures) translate into concrete spectral decompositions of the corresponding stochastic processes under study. The measures σ we consider are typically purely singular. Our proofs rely on the theory of (singular) unbounded operators in Hilbert space, and their spectral theory.
Primary 60H40, 60G15 Singular measures Secondary 60G22, 46A12 Unbounded operators Gaussian processes

Details

Metrics

Logo image