Schrödinger Bridge Over A Compact Connected Lie Group

Hamza Mahmood; Abhishek Halder; Adeel Akhtar

math.OCMar 14, 2026

Schrödinger Bridge Over A Compact Connected Lie Group

Authors: Hamza Mahmood, Abhishek Halder, Adeel Akhtar

Organizations: Department of Mechanical & Industrial Engineering at the New Jersey Institute of Technology, Newark, NJ 07102, USA · Department of Aerospace Engineering at Iowa State University, Ames, IA 50011, USA

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 https://github.com/gradslab/SbpLieGroups.git .

Figures & tables

Fig. 1: PDFs $\rho^{\mathrm{opt}}(\cdot,t)$ , $t\in[0,1]$ , for numerical example in Sec. V-A .

Fig. 2: Main plot: PDFs $\rho^{\mathrm{opt}}(\cdot,t)$ , $t\in[0,1]$ for numerical example in Sec. V-B . Inset: Sinkhorn recursion convergence for pair $(\varphi_{1},\widehat{\varphi}_{0})$ w.r.t. $d_{H}$ .

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On the Equivalence of Stochastic Control and Path Space Formulations for Schrödinger Bridges over Compact Connected Lie Groups

Hamza Mahmood, Georgiy A. Bondar, Abhishek Halder, Adeel Akhtar

Dept. of Mechanical & Industrial Engineering at the New Jersey Institute of Technology, Newark, NJ 07102, USA · Department of Applied Mathematics at the University of California at Santa Cruz, CA, 95064, USA · Dept. of Aerospace Engineering at Iowa State University, Ames, IA 50011, USA

Schrödinger Bridges Stochastic Optimal Control

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We establish the equivalence between the stochastic optimal control and path space formulations of the Schrödinger bridge problem (SBP) for the kinematic equation on a compact connected Lie group. Using the geometric concepts of horizontal lift and stochastic anti-development, we derive a Girsanov-type change-of-measure result, and show that the expected control energy equals the relative entropy of the controlled path law with respect to the reference Wiener measure. Thus, the SBP is equivalently a path space relative entropy minimization problem subject to prescribed endpoint marginals.Our result has three useful implications. From an analytic viewpoint, the shown equivalence helps prove the existence and uniqueness of the SB. From a probabilistic viewpoint, it helps interpret the SB as the most probable deviation of the uncontrolled stochastic dynamics consistent with the endpoint constraints. From a computational viewpoint, it allows using static Sinkhorn recursions to directly solve the relative entropy minimization problem and compute the optimal path measure. We illustrate the equivalence numerically on the torus

\mathbb{T}^2

. The code is publicly available at: https://github.com/gradslab/LargeDeviationSBP

Abstract

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