Information Length Analysis of Linear Autonomous Stochastic Processes

When studying the behaviour of complex dynamical systems, a statistical formulation can provide useful insights. In particular, information geometry is a promising tool for this purpose. In this paper, we investigate the information length for n-dimensional linear autonomous stochastic processes, providing a basic theoretical framework that can be applied to a large set of problems in engineering and physics. A specific application is made to a harmonically bound particle system with the natural oscillation frequency omega, subject to a damping gamma and a Gaussian white-noise. We explore how the information length depends on omega and gamma, elucidating the role of critical damping gamma=2 omega in information geometry. Furthermore, in the long time limit, we show that the information length reflects the linear geometry associated with the Gaussian statistics in a linear stochastic process.