Ornstein-Uhlenbeck Process with Fluctuating Damping

TL;DR

This paper analyzes the Ornstein-Uhlenbeck process with random damping caused by multiplicative noise, deriving explicit solutions and studying how noise affects stability and asymptotic behavior.

Contribution

It provides explicit solutions for the Ornstein-Uhlenbeck process with random damping driven by dichotomous and fractional Gaussian noise, and investigates their stability properties.

Findings

Explicit solutions for the process under different noise types

Analysis of asymptotic behavior of mean and covariance

Impact of multiplicative noise on process stability

Abstract

This paper studies Langevin equation with random damping due to multiplicative noise and its solution. Two types of multiplicative noise, namely the dichotomous noise and fractional Gaussian noise are considered. Their solutions are obtained explicitly, with the expressions of the mean and covariance determined explicitly. Properties of the mean and covariance of the Ornstein-Uhlenbeck process with random damping, in particular the asymptotic behavior, are studied. The effect of the multiplicative noise on the stability property of the resulting processes is investigated.