Stability of Ordinary Differential Equations with Colored Noise Forcing

TL;DR

This paper develops a perturbation method to analyze the stability of linear ODEs driven by colored noise, modeled as filtered white noise, using ladder operators and applied to the Mathieu equation.

Contribution

It introduces a perturbation analysis framework for moment stability of linear ODEs with colored noise forcing, extending to arbitrary linear systems and filters.

Findings

Perturbation expansion can be carried out to any order in the noise amplitude.

Method applies to a large class of linear filters and systems.

Demonstrated application to the stochastically forced Mathieu equation.

Abstract

We present a perturbation method for determining the moment stability of linear ordinary differential equations with parametric forcing by colored noise. In particular, the forcing arises from passing white noise through an $n$th order filter. We carry out a perturbation analysis based on a small parameter $\varepsilon$ that gives the amplitude of the forcing. Our perturbation analysis is based on a ladder operator approach to the vector Ornstein-Uhlenbeck process. We can carry out our perturbation expansion to any order in $\varepsilon$, for a large class linear filters, and for quite arbitrary linear systems. As an example we apply our results to the stochastically forced Mathieu equation.