Traversing the Schroedinger Bridge strait: Robert Fortet's marvelous proof redux

Abstract

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.