Most probable paths for anisotropic Brownian motions on manifolds

TL;DR

This paper characterizes the most probable paths of anisotropic Brownian motions on manifolds, linking curvature effects to path dynamics and providing numerical methods for path estimation.

Contribution

It offers a comprehensive analysis of most probable paths for anisotropic Brownian motions on manifolds, including explicit equations and numerical algorithms.

Findings

Explicit equations for most probable paths on constant curvature surfaces

Identification of curvature effects in path dynamics

Development of algorithms for mean and covariance estimation

Abstract

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.