Convergence of the Sinkhorn algorithm when the Schrödinger problem has no solution

The Sinkhorn algorithm is the most popular method for solving the Schr\"odinger problem: it is known to converge as soon as the latter has a solution, and with a linear rate when the solution has the same support as the reference coupling. Motivated by recent applications of the Schr\^odinger problem where structured stochastic processes lead to degenerate situations with possibly no solution, we show that the Sinkhorn algorithm still gives rise in this case to exactly two limit points, that can be used to compute the solution of a relaxed version of the Schr\"odinger problem, which appears as the $\Gamma$-limit of a problem where the marginal constraints are replaced by marginal penalizations. These results also allow to develop a theoretical procedure for characterizing the support of the solution - both in the original and in the relaxed problem - for any reference coupling and marginal constraints. We showcase promising numerical applications related to a model used in cell biology.