发表机构
New Jersey Institute of Technology; University of California at Santa Cruz; Iowa State University（新泽西理工学院; 加州大学圣克鲁兹分校; 爱荷华州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明紧连通李群上薛定谔桥问题的随机控制与路径空间表述等价，并给出分析、概率和计算三方面意义，数值验证于环面。
AI 中文摘要
我们建立了紧连通李群上运动学方程的薛定谔桥问题（SBP）的随机最优控制与路径空间表述之间的等价性。利用水平提升和随机反发展这一几何概念，我们推导出一个Girsanov型测度变换结果，并证明了期望控制能量等于受控路径律相对于参考维纳测度的相对熵。因此，SBP等价于在给定端点约束下的路径空间相对熵最小化问题。该结果有三个有用的意义。从分析角度看，所证明的等价性有助于证明SB的存在性和唯一性。从概率角度看，它有助于将SB解释为与端点约束一致的无控制随机动力学的最可能偏差。从计算角度看，它允许使用静态Sinkhorn递归直接求解相对熵最小化问题并计算最优路径测度。我们在环面$\mathbb{T}^2$上数值验证了该等价性。代码公开可用：此https URL
英文摘要
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
