| Summary: | 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 <i>n</i>-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 <inline-formula><math display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>, subject to a damping <inline-formula><math display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> and a Gaussian white-noise. We explore how the information length depends on <inline-formula><math display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>, elucidating the role of critical damping <inline-formula><math display="inline"><semantics><mrow><mi>γ</mi><mo>=</mo><mn>2</mn><mi>ω</mi></mrow></semantics></math></inline-formula> 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.
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