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

  • Yongxin Chen
  • Tryphon T. Georgiou
  • Michele Pavon


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.


Optimal transport Schrödinger bridge Stochastic control 

Mathematics Subject Classification

60J60 49L20 49J20 35Q35 28A50 



Research partially supported by the NSF under Grant ECCS-1509387, the AFOSR under Grants FA9550-12-1-0319, and FA9550-15-1-0045 and by the University of Padova Research Project CPDA 140897. Part of the research of M.P. was conducted during a stay at the Courant Institute of Mathematical Sciences of the New York University whose hospitality is gratefully acknowledged.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Yongxin Chen
    • 1
  • Tryphon T. Georgiou
    • 1
  • Michele Pavon
    • 2
  1. 1.University of MinnesotaMinneapolisUSA
  2. 2.Università di PadovaPadovaItaly

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