  In the early 1930's, Erwin Schroedinger, motivated by his quest for a more
classical formulation of quantum mechanics, posed a large deviation problem for
a cloud of independent Brownian particles. He showed that the solution to the
problem could be obtained trough a system of two linear equations with
nonlinear coupling at the boundary (Schr\"odinger system). Existence and
uniqueness for such a system, which represents a sort of bottleneck for the
problem, was first established by R. Fortet in 1938/40 under rather general
assumptions by proving convergence of an ingenious but complex approximation
method. It is the first proof of what are nowadays called Sinkhorn-type
algorithms in the much more challenging continuous case. Schr\"odinger bridges
are also an early example of the maximum entropy approach and have been more
recently recognized as a regularization of the important Optimal Mass Transport
problem. Unfortunately, Fortet's contribution is by and large ignored in
contemporary literature. This is likely due to the complexity of his approach
coupled with an idiosyncratic exposition style and to missing details and steps
in the proofs. Nevertheless, Fortet's approach maintains its importance to this
day as it provides the only existing algorithmic proof under rather mild
assumptions. It can be adapted, in principle, to other relevant problems such
as the regularized Wasserstein barycenter problem. It is the purpose of this
paper to remedy this situation by rewriting the bulk of his paper with all the
missing passages and in a transparent fashion so as to make it fully available
to the scientific community. We consider the problem in $R^d$ rather than $R$
and use as much as possible his notation to facilitate comparison.
