In this paper, we explore quantitative stability of multi-marginal Schrödinger bridges with respect to the marginal constraints. We focus on the case where the number of marginal constraints is large (i.e. ``many-marginals"). When this number increases, we show that the Kullback--Leibler (KL) divergence between two multi-marginal Schrödinger bridges, as measures on the path space, can be asymptotically bounded by the terminal marginal KL divergence and a time-integrated squared discrepancy {that combines} Wasserstein-2 geodesic velocity fields with a log-density gradient term. Our stability upper bound is also asymptotically tight: it converges to zero as the number of marginal constraints increases with unperturbed marginal constraints. To the best of our knowledge, this is the first such stability result that addresses the many-marginal regime, giving error estimates that are asymptotically independent of the number of marginals. To achieve our result, the key step is to derive an asymptotic expansion (of order $k\ge 2$) of Schrödinger potentials with respect to a diminishing regularization coefficient. This result can also be applied to deriving asymptotic expansions of entropic Brenier maps in entropic optimal self-transport problems. As byproducts of our analyses, we also establish the asymptotic expansion of entropic optimal transport cost with respect to the diminishing regularization coefficient when two marginal constraints are sufficiently close. We also prove a stability property of the Schrödinger functional.

翻译：本文研究了多边际薛定谔桥关于边际约束的定量稳定性。我们重点关注边际约束数量较大（即"多边际"）的情形。当边际数目增加时，我们证明了作为路径空间上测度的两个多边际薛定谔桥之间的库尔贝克-莱布勒散度可渐近地由终端边际KL散度与一个包含Wasserstein-2测地速度场和对数密度梯度项的时间积分平方偏差共同界定。我们的稳定性上界也是渐近紧的：当边际约束未受扰动且数目增加时，该上界趋近于零。据我们所知，这是首个针对多边际体制的稳定性结果，其误差估计渐近独立于边际数量。为实现该结果的关键步骤，我们推导了薛定谔势关于衰减正则化系数的（$k\ge 2$阶）渐近展开。该结果还可应用于熵最优自输运问题中熵Brenier映射的渐近展开。作为分析的副产品，我们还建立了当两个边际约束足够接近时，熵最优输运代价关于衰减正则化系数的渐近展开，并证明了薛定谔泛函的稳定性性质。