We develop a novel stability theory for Sinkhorn semigroups based on Lyapunov techniques and quantitative contraction coefficients, and establish exponential convergence of Sinkhorn iterations on weighted Banach spaces. This operator-theoretic framework yields explicit exponential decay rates of Sinkhorn iterates toward Schrödinger bridges with respect to a broad class of $φ$-divergences and Kantorovich-type distances, including relative entropy, squared Hellinger integrals, $α$-divergences, weighted total variation norms, and Wasserstein distances. To the best of our knowledge, these results provide the first systematic contraction inequalities of this kind for entropic transport and the Sinkhorn algorithm. We further introduce Lyapunov contraction principles under minimal regularity assumptions, leading to quantitative exponential stability estimates for a large family of Sinkhorn semigroups. The framework applies to models with polynomially growing potentials and heavy-tailed marginals on general normed spaces, as well as to more structured boundary state-space models, including semicircle transitions and Beta, Weibull, and exponential marginals, together with semi-compact settings. Finally, our approach extends naturally to statistical finite mixtures of such models, including kernel-based density estimators arising in modern generative modeling.

翻译：基于Lyapunov技术与定量收缩系数，我们为Sinkhorn半群建立了一套新颖的稳定性理论，并在加权Banach空间上证明了Sinkhorn迭代的指数收敛性。该算子理论框架针对一大类φ-散度与Kantorovich型距离（包括相对熵、平方Hellinger积分、α-散度、加权全变差范数以及Wasserstein距离），给出了Sinkhorn迭代向薛定谔桥收敛的显式指数衰减率。据我们所知，这些结果为熵传输及Sinkhorn算法提供了首个此类系统性的收缩不等式。我们进一步在最小正则性假设下引入Lyapunov收缩原理，从而为一大族Sinkhorn半群导出定量的指数稳定性估计。该框架适用于一般赋范空间上具有多项式增长势函数与重尾边缘分布的模型，也适用于更具结构性的边界状态空间模型（包括半圆转移与Beta分布、威布尔分布及指数分布边缘分布），以及半紧致设定。最后，我们的方法可自然推广至此类模型的统计有限混合情形，包括现代生成建模中出现的基于核的密度估计器。