  This paper exploit the equivalence between the Schr\"odinger Bridge problem
and the entropy penalized optimal transport in order to find a different
approach to the duality, in the spirit of optimal transport. This approach
results in a priori estimates which are consistent in the limit when the
regularization parameter goes to zero. In particular, we find a new proof of
the existence of maximizing entropic-potentials and therefore, the existence of
a solution of the Schr\"odinger system. Our method extends also when we have
more than two marginals: we can provide an alternative proof of the convergence
of the Sinkhorn algorithm with two marginals and we show that the Sinkhorn
algorithm converges in the multi-marginal case.
