  We study the stability of entropically regularized optimal transport with
respect to the marginals. Given marginals converging weakly, we establish a
strong convergence for the Schr\"odinger potentials describing the density of
the optimal couplings. When the marginals converge in total variation, the
optimal couplings also converge in total variation. This is applied to show
that Sinkhorn's algorithm converges in total variation when costs are quadratic
and marginals are subgaussian, or more generally for all continuous costs
satisfying an integrability condition.

Probability;Analysis of PDEs;Functional Analysis;Optimization and Control