Most probable paths for developed processes

TL;DR

This paper develops a unified framework for identifying most probable paths of a broad class of manifold-valued stochastic processes, extending beyond classical cases to include processes with complex structures.

Contribution

It introduces a general approach using the Onsager-Machlup function on the anti-development of processes, providing explicit equations for development most probable paths.

Findings

Derived explicit equations for development most probable paths.

Extended the framework to processes on Lie groups and shape spaces.

Applied the method to various stochastic processes in geometric contexts.

Abstract

Optimal paths for the classical Onsager-Machlup function 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 function on the anti-development of such processes. The construction encompasses large classes of manifold-valued process and results in explicit equation systems for the paths that we denote \emph{development 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.