Optimal paths for the classical Onsager-Machlup functional determining most probable paths between points on a manifold are only explicitly identified for specific processes, for example the Riemannian Brownian motion. This leaves out large classes of manifold-valued processes such as processes with parallel transported non-trivial diffusion matrix, processes with rank-deficient generator and sub-Riemannian processes, and push-forwards to quotient spaces. In this paper, we construct a general approach to definition and identification of most probable paths by measuring the Onsager-Machlup functional on the anti-developement of such processes. The construction encompasses large classes of manifold-valued process and results in explicit equation systems for most probable paths. We define and derive these results and apply them to several cases of stochastic processes on Lie groups, homogeneous spaces, and landmark spaces appearing in shape analysis.
翻译:经典 Onsager-Machlup 功能最优化的路径, 确定一个方块上最可能的路径之间的最可能路径, 只是为特定过程而明确确定, 例如 Riemannian Brownian 运动 。 这遗漏了大量的多种有价值过程, 如平行运输的非三边扩散矩阵的过程、 级低生成器和亚里曼尼进程, 以及推向商数空间的过程 。 在本文中, 我们通过测量 Onsager- Machlup 功能在反开发过程中的功能, 构建了一种通用的办法来定义和识别最可能路径 。 构建包含大量多层有价值的过程, 以及针对大多数可能路径的直方程系统的结果 。 我们定义和生成这些结果, 并将其应用到Lie 群、 同一空间和形状分析中出现的里程碑空间的数个随机过程中 。