We study stochastic density control between Gaussian-mixture endpoint distributions under Brownian prior dynamics. Since the direct Schrödinger bridge between Gaussian mixtures is generally not available in closed form, we introduce a lifted path-space construction in which each trajectory is augmented with a source--target component label. Consequently, the problem decomposes into Gaussian component-to-component Schrödinger bridges with explicit marginal, drift, and cost formulas, while the mixture-level assignment reduces to a finite-dimensional entropic coupling problem with a Sinkhorn scaling form. We then analyze the projection obtained by discarding or forgetting the label. By construction, the projected law satisfies the original Gaussian-mixture endpoint constraints, but its relative entropy generally differs from the lifted relative entropy by a nonnegative conditional label-information gap. This gap reveals a path-space obstruction: the lifted optimizer cannot, in general, be identified with the direct unlabeled Schrödinger bridge after projection. We also derive the posterior-averaged Markov drift associated with the projected marginal flow, prove a kinetic-energy upper bound, and identify a common path-potential condition under which the projection gap vanishes. Several numerical illustrations showing density and shape control are recorded for a self-contained exposition.

翻译：我们研究在布朗先验动力学下高斯混合端点分布之间的随机密度控制问题。由于高斯混合间的直接薛定谔桥通常无法以闭式形式获得，我们引入一种提升路径空间构造，为每条轨迹附加一个源-目标分量标签。由此，该问题可分解为具有显式边际、漂移和代价公式的高斯分量间薛定谔桥，而混合级分配则简化为具有Sinkhorn缩放形式的有限维熵耦合问题。随后我们分析通过丢弃或遗忘标签所得的投影。根据构造，投影分布满足原始高斯混合端点约束，但其相对熵通常与提升相对熵相差一个非负的条件标签信息间隙。此间隙揭示了一个路径空间障碍：提升优化器通常无法在投影后等同于直接的无标签薛定谔桥。我们还推导了与投影边际流相关联的后验平均马尔可夫漂移，证明了动能上限，并识别出投影间隙消失的公共路径势条件。为达到自包含阐述，本文记录了若干展示密度与形状控制的数值实例。