Linear State space theory in the white noise space setting

TL;DR

This paper develops a theoretical framework for analyzing linear state space systems within the white noise space, emphasizing the role of commutative rings of power series and the deterministic part of stochastic systems.

Contribution

It introduces a novel approach to characterize stochastic systems using commutative rings and links system properties to their deterministic averages.

Findings

Transfer functions are rational functions with coefficients in a commutative ring.

System characteristics are determined by the deterministic average behavior.

The approach provides new insights into stochastic system analysis.

Abstract

We study state space equations within the white noise space setting. A commutative ring of power series in a countable number of variables plays an important role. Transfer functions are rational functions with coefficients in this commutative ring, and are characterized in a number of ways. A major feature in our approach is the observation that key characteristics of a linear, time invariant, stochastic system are determined by the corresponding characteristics associated with the deterministic part of the system, namely its average behavior.