  The Sinkhorn algorithm is the most popular method for solving the entropy
minimization problem called the Schr\"odinger problem: in the non-degenerate
cases, the latter admits a unique solution towards which the algorithm
converges linearly. Here, motivated by recent applications of the Schr\"odinger
problem with respect to structured stochastic processes (such as increasing
ones), we study the Sinkhorn algorithm in degenerate cases where it might
happen that no solution exist at all. We show that in this case, the algorithm
ultimately alternates between two limit points. Moreover, these limit points
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 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, giving
its typical shape and designing a procedure to compute it quickly. We showcase
promising numerical applications related to a model used in cell biology.
