Generalized Fock space and moments

TL;DR

This paper extends the theory of generalized stochastic processes to broader probability spaces, including grey noise, by developing a moment-based framework and emphasizing the role of strong algebras.

Contribution

It introduces a new framework for stochastic processes in generalized spaces, recasting the Wiener-Itô expansion as a moment problem and highlighting strong algebras.

Findings

Explicit calculation of moments for generalized processes

Demonstration of the framework's applicability to stochastic process analysis

Extension of white noise space theory to grey noise and other spaces

Abstract

In this paper we develop a framework to extend the theory of generalized stochastic processes in the Hida white noise space to more general probability spaces which include the grey noise space. To obtain a Wiener-It\^o expansion we recast it as a moment problem and calculate the moments explicitly. We further show the importance of a family of topological algebras called strong algebras in this context. Furthermore we show the applicability of our approach to the study of stochastic processes.