发表机构
Universidade Federal do Rio de Janeiro; University of Tartu（里约热内卢联邦大学; 塔尔图大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究建立高斯流形α-连接测地线与连续随机过程的一一对应，构造广义桥过程，其恢复力决定连接参数α，并证明慢驱动最优协议对应非度量测地线。
AI 中文摘要
我们建立了高斯统计流形上与α-连接单参数族相关的测地线与一类由时间无关噪声强度刻画的连续随机过程之间的一一对应关系。我们证明了期望参数中的测地线自然地分为三个不同的几何类别，其中边界连接类使我们能够构造具有恒定扩散的显式线性随机实现，代表一种广义桥过程。这一结果展示了该高斯类中的连续随机过程如何沿几何曲线扩展。在适当的操作极限下，该广义桥过程简化为基本的随机动力学，即奥恩斯坦-乌伦贝克（OU）弛豫或自由布朗扩散。关键在于，控制所得OU弛豫的物理恢复力直接决定了底层连接参数α，为约束流形几何提供了具体的物理可观测量。根据所选的仿射连接表示，该恢复力可归因于标量曲率或纯粹的非度量性，从而与引力的几何三位一体建立了直接的概念类比。此外，将该框架应用于驱动随机热力学，我们证明了慢驱动极限下功最小化的最优协议恰好与配备$(g, {}^{(1/2)}\Gamma, {}^{(-1/2)}\Gamma)$的非度量统计流形的期望测地线重合，突显了非度量性在信息几何中的积极物理作用。
英文摘要
We establish a one-to-one correspondence between geodesics associated with the one-parameter family of $α$-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 $α$, 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 $(g, {}^{(1/2)}Γ, {}^{(-1/2)}Γ)$, highlighting the active physical role of non-metricity in information geometry.
Comments20 pages, 2 figures
