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The Sinkhorn algorithm is the most popular method for solving the entropyminimization problem called the Schr\"odinger problem: in the non-degeneratecases, the latter admits a unique solution towards which the algorithmconverges linearly. Here, motivated by recent applications of the Schr\"odingerproblem with respect to structured stochastic processes (such as increasingones), we study the Sinkhorn algorithm in degenerate cases where it mighthappen that no solution exist at all. We show that in this case, the algorithmultimately alternates between two limit points. Moreover, these limit pointscan be used to compute the solution of a relaxed version of the Schr\"odingerproblem, which appears as the $\Gamma$-limit of a problem where the marginalconstraints are replaced by asymptotically large marginal penalizations,exactly in the spirit of the so-called unbalanced optimal transport. Finally,our work focuses on the support of the solution of the relaxed problem, givingits typical shape and designing a procedure to compute it quickly. We showcasepromising numerical applications related to a model used in cell biology.

Convergence of the Sinkhorn algorithm when the Schr\"odinger problem has no solution