Symplectic techniques for stochastic differential equations on reductive Lie groups with applications to Langevin diffusions
Abstract
We show how Langevin diffusions can be interpreted in the context of stochastic Hamiltonian systems with structure-preserving noise and dissipation on reductive Lie groups. Reductive Lie groups provide the setting in which the Lie group structure is compatible with Riemannian structures, via the existence of bi-invariant metrics. This structure allows for the explicit construction of Riemannian Brownian motion via symplectic techniques, which permits the study of Langevin diffusions with noise in the position coordinate as well as Langevin diffusions with noise in both momentum and position.
Cite
@article{arxiv.2504.02707,
title = {Symplectic techniques for stochastic differential equations on reductive Lie groups with applications to Langevin diffusions},
author = {Erwin Luesink and Oliver D. Street},
journal= {arXiv preprint arXiv:2504.02707},
year = {2025}
}
Comments
29 pages, fourth version. We made the Stratonovich integration that is necessary to construct the Riemannian Brownian motion more clear and made the Stratonovich time-ordered exponential explicit to emphasise the noncommutativity. All comments are welcome!