A potpourri of results in the general theory of stochastic noises

Abstract

The objects under inspection, on a given probability space, are noise(-type) Boolean algebras -- distributive non-empty sublattices of the lattice of all complete sub-$\sigma$-fields, whose every element admits an independent complement. Special attention is given to the spectral decompositions of the algebras of operators generated by the conditional expectations of their members (acting on $\mathrm{L}^2$). Atoms of the spectra are identified in explicit terms. For a reverse filtration admitting an innovating sequence of equiprobable random signs, a discreteness property of the spectral measure of the associated noise Boolean algebra is shown to imply product-typeness. Noise projections on the spectral space are introduced, which correspond to restricting a noise to a part of its domain space. They appear to play a natural (albeit technical) r\^ole in the general analysis. In particular through them manifests itself the tensor structure of the spectral decomposition. The spectrum is precisely delineated in the classical case (i.e. when the noise Boolean algebra is complete), a kind-of standard "symmetric Fock space" form thereof is procured. The latter result leads to a new characterization of classicality and blackness involving "spectral independence".