A Contraction Theory for Sinkhorn and Schrodinger Bridges via Log-Sobolev Inequalities

We develop a quantitative contraction framework for Schrodinger and Sinkhorn bridges based on transportation-cost inequalities and Riccati matrix difference equations. Our approach combines logarithmic Sobolev and Talagrand-type inequalities to obtain explicit entropy and Wasserstein contraction bounds for Sinkhorn bridge measures, entropic optimal transport plans, and the associated Markov transport maps. A key feature of the analysis is the interplay between transport-cost inequalities and matrix Riccati difference equations arising in filtering and stochastic control. The results are established under local regularity assumptions on the reference transition, formulated in terms of curvature, Lipschitz continuity, and Fisher-information bounds. Within this general setting, we derive quantitative stability and convergence estimates for Schrodinger bridges and Sinkhorn iterates that are robust with respect to the choice of reference measure. As a main application, we specialize the theory to linear-Gaussian reference transitions, where the Gaussian structure permits sharp constants, refined exponential decay rates, and continuity estimates for Schrodinger bridges, Sinkhorn iterates, barycentric projections, conditional covariances, and proximal sampler semigroups. In this setting, we recover and extend several known contraction results for entropic and Wasserstein distances, and obtain new quantitative bounds that improve previously available rates. Our results provide a unified probabilistic framework for stability, regularity, and convergence of Sinkhorn algorithms.