AI 中文总结
研究次黎曼流形上反射薛定谔桥问题，针对几何和退化性困难，引入内在斜反射机制，证明相关过程有光滑正转移密度，通过正反向偏微分方程系统及基于偏微分方程的Sinkhorn迭代求解最优控制并举例说明。
AI 中文摘要
我们阐述并解决了反射次黎曼薛定谔桥问题：在有退化扩散和硬状态约束的欠驱动动力学下，有界域上概率分布的最小能量传输。主要困难在于几何和退化性：欧几里得法向反射通常与水平子丛不兼容，参考测度可能没有完全支撑。我们通过引入与次黎曼结构兼容的内在斜反射机制来解决这些问题。在霍尔曼德和非特征边界假设下，证明了反射退化参考过程具有光滑、严格正的转移密度。最优控制由具有非对称边界条件的正反向偏微分方程系统表征。由于通常无法得到显式转移密度，我们开发了一种基于偏微分方程的Sinkhorn迭代来直接施加这些边界条件，并举例说明了该框架。
英文摘要
We formulate and solve the reflected sub-Riemannian Schrödinger bridge (SB) problem: minimum-energy transport of probability distributions on a bounded domain for underactuated dynamics with degenerate diffusion and hard state constraints. The main difficulties are geometric and degeneracy: Euclidean normal reflection is generally incompatible with the horizontal subbundle, since it may push the process in directions not generated by the admissible control and noise fields. Also, the reference measure may not have full support. We address these by introducing an intrinsic oblique reflection mechanism that is compatible to the sub-Riemannian structure. Under Hörmander and non-characteristic boundary assumptions, we prove that the reflected degenerate reference process admits a smooth, strictly positive transition density. The resulting optimal control is characterized by a forward--backward PDE system with asymmetric boundary conditions: an oblique Neumann condition for the backward factor and a normal no-flux condition for the forward factor. Since explicit transition densities are generally not available in this setting, we develop a PDE-based Sinkhorn iteration that enforces these boundary conditions directly. Our examples reveal that SB depends on the topology of the domain and the geometry of the diffusion.
