Stochastic Processes as Non-Metric Geodesics in Information Geometry

Abstract

We establish a one-to-one correspondence between geodesics associated with the one-parameter family of $\alpha$-connections on the Gaussian statistical manifold and a class of continuous stochastic processes characterized by a time-independent noise intensity. We demonstrate that geodesics in expectation parameters naturally classify into three distinct geometric categories, among which the Boundary-connecting class allows us to construct an explicit linear stochastic realization with constant diffusion, representing a generalized bridge process. This result demonstrates how continuous stochastic processes within this Gaussian class can be extended along geometric curves. Under appropriate operational limits, this generalized bridge process reduces to fundamental stochastic dynamics, either Ornstein-Uhlenbeck (OU) relaxation or free Brownian diffusion. Crucially, the physical restoring force governing the resulting OU relaxation directly determines the underlying connection parameter $\alpha$, providing a concrete physical observable to constrain the manifold geometry. Depending on the chosen affine connection representation, this restoring force can be attributed either to scalar curvature or purely to non-metricity, establishing a direct conceptual analogy with the Geometrical Trinity of Gravity. Furthermore, applying this framework to driven stochastic thermodynamics, we show that the work-minimizing optimal protocol in the slow-driving limit coincides precisely with an expectation geodesic of the statistical manifold with non-metricity equipped with $(g, {}^{(1/2)}\Gamma, {}^{(-1/2)}\Gamma)$, highlighting the active physical role of non-metricity in information geometry.