TY - JOUR

T1 - Most Probable Paths for Anisotropic Brownian Motions on Manifolds

AU - Grong, Erlend

AU - Sommer, Stefan

PY - 2024

Y1 - 2024

N2 - 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.

AB - 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.

U2 - 10.1007/s10208-022-09594-4

DO - 10.1007/s10208-022-09594-4

M3 - Journal article

VL - 24

SP - 313

EP - 345

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

SN - 1615-3375

ER -