Linear stochastic systems: a white noise approach

TL;DR

This paper introduces a novel white noise-based framework for analyzing linear stochastic systems, incorporating randomness into transfer functions and establishing various stability theorems for both discrete and continuous time systems.

Contribution

It develops a new approach using white noise analysis tools to study stability of stochastic systems with random transfer functions, extending classical results.

Findings

Proves BIBO stability theorems for stochastic systems

Establishes dissipative system stability in both discrete and continuous time

Analyzes $ ext{l}_1$-$ ext{l}_2$ and ${ extbf{L}}_2$-${ extbf{L}}_ extinfty$ stability conditions

Abstract

Using the white noise setting, in particular the Wick product, the Hermite transform, and the Kondratiev space, we present a new approach to study linear stochastic systems, where randomness is also included in the transfer function. We prove BIBO type stability theorems for these systems, both in the discrete and continuous time cases. We also consider the case of dissipative systems for both discrete and continuous time systems. We further study $\ell_1$-$\ell_2$ stability in the discrete time case, and ${\mathbf L}_2$-${\mathbf L}_\infty$ stability in the continuous time case.