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On the Relation Between Optimal Transport and Schrödinger Bridges: A Stochastic Control Viewpoint

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  • Yongxin Chen

    (University of Minnesota)

  • Tryphon T. Georgiou

    (University of Minnesota)

  • Michele Pavon

    (Università di Padova)

Abstract

We take a new look at the relation between the optimal transport problem and the Schrödinger bridge problem from a stochastic control perspective. Our aim is to highlight new connections between the two that are richer and deeper than those previously described in the literature. We begin with an elementary derivation of the Benamou–Brenier fluid dynamic version of the optimal transport problem and provide, in parallel, a new fluid dynamic version of the Schrödinger bridge problem. We observe that the latter establishes an important connection with optimal transport without zero-noise limits and solves a question posed by Eric Carlen in 2006. Indeed, the two variational problems differ by a Fisher information functional. We motivate and consider a generalization of optimal mass transport in the form of a (fluid dynamic) problem of optimal transport with prior. This can be seen as the zero-noise limit of Schrödinger bridges when the prior is any Markovian evolution. We finally specialize to the Gaussian case and derive an explicit computational theory based on matrix Riccati differential equations. A numerical example involving Brownian particles is also provided.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:169:y:2016:i:2:d:10.1007_s10957-015-0803-z
    DOI: 10.1007/s10957-015-0803-z
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    References listed on IDEAS

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    1. R. Filliger & M.-O. Hongler & L. Streit, 2008. "Connection between an Exactly Solvable Stochastic Optimal Control Problem and a Nonlinear Reaction-Diffusion Equation," Journal of Optimization Theory and Applications, Springer, vol. 137(3), pages 497-505, June.
    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.
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    Cited by:

    1. Yongxin Chen & Tryphon T. Georgiou & Michele Pavon, 2018. "Steering the Distribution of Agents in Mean-Field Games System," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 332-357, October.
    2. 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.
    3. Gentil, Ivan & Léonard, Christian & Ripani, Luigia & Tamanini, Luca, 2020. "An entropic interpolation proof of the HWI inequality," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 907-923.
    4. Conforti, Giovanni & Léonard, Christian, 2022. "Time reversal of Markov processes with jumps under a finite entropy condition," Stochastic Processes and their Applications, Elsevier, vol. 144(C), pages 85-124.

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