Brownian motion on manifolds with non-trivial diffusion coefficient can be constructed by stochastic development of Euclidean Brownian motions using the fiber bundle of linear frames. We provide a comprehensive study of paths for such processes that are most probable in the sense of Onsager-Machlup, however with path probability measured on the driving Euclidean processes. We obtain both a full characterization of the resulting family of most probable paths, reduced equation systems for the path dynamics where the effect of curvature is directly identifiable, and explicit equations in special cases, including constant curvature surfaces where the coupling between curvature and covariance can be explicitly identified in the dynamics. We show how the resulting systems can be integrated numerically and use this to provide examples of most probable paths on different geometries and new algorithms for estimation of mean and infinitesimal covariance.

翻译：使用线性框架的纤维捆绑,对Euclidean Brownian运动进行随机发展,可以构建在非三角扩散系数的方块上的布朗运动。我们全面研究Onsager-Machlup 意义上最有可能的这些过程的路径,但在驱动 Euclidean 过程上测量了路径概率。我们获得了对由此形成的最可能路径的完整描述,在曲线效果可以直接识别的路径动态方面减少了方程式系统,在特殊情况下也采用了明确的方程式,包括恒定的曲线表面,在动态中可以明确辨明曲线和共变形之间的组合。我们展示了由此产生的系统如何以数字方式整合,并以此为不同地理特征和估计平均和无限共性的新算法提供最可能路径的实例。