[2603.14049] Schrödinger Bridge Over A Compact Connected Lie Group

Mathematics > Optimization and Control arXiv:2603.14049 (math) [Submitted on 14 Mar 2026 (v1), last revised 20 Mar 2026 (this version, v2)] Title:Schrödinger Bridge Over A Compact Connected Lie Group Authors:Hamza Mahmood, Abhishek Halder, Adeel Akhtar View a PDF of the paper titled Schr\"odinger Bridge Over A Compact Connected Lie Group, by Hamza Mahmood and 2 other authors View PDF HTML (experimental) Abstract:This work studies the Schrödinger bridge problem for the kinematic equation on a compact connected Lie group. The objective is to steer a controlled diffusion between given initial and terminal densities supported over the Lie group while minimizing the control effort. We develop a coordinate-free formulation of this stochastic optimal control problem that respects the underlying geometric structure of the Lie group, thereby avoiding limitations associated with local parameterizations or embeddings in Euclidean spaces. We establish the existence and uniqueness of solution to the corresponding Schrödinger system. Our results are constructive in that they derive a geometric controller that optimally interpolates probability densities supported over the Lie group. To illustrate the results, we provide numerical examples on $\mathsf{SO}(2)$ and $\mathsf{SO}(3)$. The codes and animations are publicly available at this https URL . Subjects: Optimization and Control (math.OC); Machine Learning (cs.LG); Systems and Control (eess.SY); Probability (math.PR) Cite as: arXiv:26...