Coordinate-transformation approach for contractivity of kinetic Langevin on curved spaces

Determine whether and how a coordinate-transformation technique analogous to the constant linear transformation used for proving contractivity of kinetic Langevin in Euclidean spaces can be generalized to curved spaces such as compact Lie groups, in a manner that respects manifold geometry and yields rigorous convergence guarantees.

Background

In Euclidean settings with strongly convex potentials, contractivity and convergence of kinetic Langevin dynamics can be shown after an appropriate constant coordinate transformation, as in prior work by Dalalyan and Riou-Durand. The authors point out that on curved spaces, such a transformation cannot remain constant due to geometric nonlinearity, and may fail to induce a compatible metric, making the approach unclear.

Because compact Lie groups lack nontrivial geodesically convex functions, even replicating convexity-based analyses is vacuous; thus clarifying the role of coordinate transformations in this setting is a meaningful and unresolved problem.