Stochastic processes induced by singular operators

TL;DR

This paper explores a broad class of Gaussian stochastic processes driven by singular measures, linking measure-theoretic properties to spectral decompositions using operator theory in Hilbert spaces.

Contribution

It extends previous work by analyzing processes associated with purely singular measures and connecting measure equivalence classes to spectral properties.

Findings

Spectral decompositions correspond to measure equivalence classes.

Singular measures lead to distinct spectral behaviors.

Operator theory provides the framework for analysis.

Abstract

In this paper we study a general family of multivariable Gaussian stochastic processes. Each process is prescribed by a fixed Borel measure $\sigma$ on $\mathbb R^n$. The case when $\sigma$ 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 $\sigma$ we consider are typically purely singular. Our proofs rely on the theory of (singular) unbounded operators in Hilbert space, and their spectral theory.