White Noise Space Analysis and Multiplicative Change of Measures

TL;DR

This paper constructs a family of Gaussian processes with explicit transforms, demonstrating that their path-space measures are mutually singular and linking these to pairwise disjoint CCR representations using advanced infinite-dimensional analysis.

Contribution

It introduces explicit formulas for transforms of Gaussian processes and establishes their mutual singularity and inequivalence of associated CCR representations.

Findings

Explicit formulas for Gaussian process transforms

Mutual singularity of path-space measures

Disjoint CCR representations corresponding to different processes

Abstract

In this paper we display a family of Gaussian processes, with explicit formulas and transforms. This is presented with the use of duality tools in such a way that the corresponding path-space measures are mutually singular. We make use of a corresponding family of representations of the canonical commutation relations (CCR) in an infinite number of degrees of freedom. A key feature of our construction is explicit formulas for associated transforms; these are infinite-dimensional analogues of Fourier transforms. Our framework is that of Gaussian Hilbert spaces, reproducing kernel Hilbert spaces, and Fock spaces. The latter forms the setting for our CCR representations. We further show, with the use of representation theory, and infinite-dimensional analysis, that our pairwise inequivalent probability spaces (for the Gaussian processes) correspond in an explicit manner to pairwise disjoint CCR representations.