We discuss entropically regularized optimal transport and its stability with respect to the marginals. A qualitative result (for weak convergence) is obtained using the geometric notion of c-cyclical monotonicity and a quantitative result (for Wasserstein distance) is obtained by control theoretic methods. These results can be applied to deduce convergence of Sinkhorn’s algorithm for unbounded cost functions such as the quadratic cost and find a convergence rate in Wasserstein sense. Based on joint works with Espen Bernton, Stephan Eckstein, Promit Ghosal, Johannes Wiesel.

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