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

arXiv: https://arxiv.org/abs/1805.02695 · HTML: https://mlsanity.com/paper/1805.02695/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 May 2018.

## Built on by (most cited first, 6 of 6)

- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Soft-constrained Schrodinger Bridge: a Stochastic Control Approach](https://mlsanity.com/paper/2403.01717/) (2024)
- [Data Assimilation for Sign-indefinite Priors: A generalization of Sinkhorn's algorithm](https://mlsanity.com/paper/2308.11791/) (2023)
- [New Trends in the Stability of Sinkhorn Semigroups](https://mlsanity.com/paper/2601.12633/) (2026)
- [Mirror Bridges Between Probability Measures](https://mlsanity.com/paper/2410.07003/) (2024)
- [Exact Solutions to the Quantum Schr"odinger Bridge Problem](https://mlsanity.com/paper/2509.25980/) (2025)

## Abstract (arXiv)

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 Rd rather than R and use as much as possible his notation to facilitate comparison.