  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.
