arXiv:2503.09887v2 Announce Type: replace-cross Abstract: We develop a novel semigroup stability analysis based on Lyapunov techniques and contraction coefficients to prove exponential convergence of Sinkhorn equations on weighted Banach spaces. This operator-theoretic framework yields exponential decays of Sinkhorn iterates towards Schr\"odinger bridges with respect to general classes of $\phi$-divergences and Kantorovich-type criteria, including the relative entropy, squared Hellinger integrals, $\alpha$-divergences as well as weighted total variation norms and Wasserstein distances. To the best of our knowledge, these contraction inequalities are the first results of this type in the literature on entropic transport and the Sinkhorn algorithm. We also provide Lyapunov contractions principles under minimal regularity conditions that allow to provide quantitative exponential stability estimates for a large class of Sinkhorn semigroups. We apply this novel framework in a variety of situations, ranging from polynomial growth potentials and heavy tailed marginals on general normed spaces to more sophisticated boundary state space models, including semi-circle transitions, Beta, Weibull, exponential marginals as well as semi-compact models. Last but not least, our approach also allows to consider statistical finite mixture of the above models, including kernel-type density estimators of complex data distributions arising in generative modeling.