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.

References

In Euclidean space, when U is strongly convex, one can prove the convergence of kinetic Langevin by showing the contractivity of dynamics after a constant coordinate transformation \citep{dalalyan2020sampling}. It is unclear to us how to do the coordinate transformation in curved spaces, because it can no longer be constant due to the nonlinearity of space, but then additional challenges arise (e.g., it no longer induces a metric as essentially done in \citep{dalalyan2020sampling}).

— Convergence of Kinetic Langevin Monte Carlo on Lie groups  (2403.12012 - Kong et al., 2024) in Section "More details about related works", Subsection "Discussion about commonly used conditions on potential functions"

Finally, it is intriguing to construct a kinetic theory for a closed system consisting of particles with a non-isotropic kinetic energy term, namely when the bath itself has a curl-force Hamiltonian. Could such a system relax strongly (i.e. is the corresponding Markov semigroup a contraction), and if so, would it relax to the corresponding equilibrium or a nonequilibrium steady state?

— Hamiltonian curl-forces in systems coupled to multiple thermal reservoirs  (2610.08423 - Chor et al., 6 Oct 2026) in Discussion and outlook