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Traversing the Schrödinger Bridge Strait: Robert Fortet’s Marvelous Proof Redux

Author

Listed:
  • Montacer Essid

    (New York University)

  • Michele Pavon

    (Università di Padova)

Abstract

In the early 1930s, Erwin Schrödinger, 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 through a system of two linear equations with nonlinear coupling at the boundary (Schrödinger system). Existence and uniqueness for such a system, which represents a sort of bottleneck for the problem, was first established by Fortet in 1938/1940 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. Schrödinger 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 due 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, in the continuous setting, under rather mild assumptions. It can be adapted, in principle, to other relevant optimal transport problems. 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 $${\mathbb {R}}^d$$ R d rather than in $${\mathbb {R}}$$ R and use as much as possible his notation to facilitate comparison.

Suggested Citation

  • Montacer Essid & Michele Pavon, 2019. "Traversing the Schrödinger Bridge Strait: Robert Fortet’s Marvelous Proof Redux," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 23-60, April.
  • Handle: RePEc:spr:joptap:v:181:y:2019:i:1:d:10.1007_s10957-018-1436-9
    DOI: 10.1007/s10957-018-1436-9
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    References listed on IDEAS

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    1. Alfred Galichon & Scott Kominers & Simon Weber, 2015. "The Nonlinear Bernstein-Schrödinger Equation in Economics," SciencePo Working papers Main hal-03393000, HAL.
    2. Mikami, Toshio & Thieullen, Michèle, 2006. "Duality theorem for the stochastic optimal control problem," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1815-1835, December.
    3. Yongxin Chen & Tryphon T. Georgiou & Michele Pavon, 2016. "On the Relation Between Optimal Transport and Schrödinger Bridges: A Stochastic Control Viewpoint," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 671-691, May.
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