Skip to main content
Springer Nature Link
Log in

Extremal Curves in Wasserstein Space

  • Conference paper
  • First Online:
Geometric Science of Information (GSI 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10589))

Included in the following conference series:

  • 2407 Accesses

Abstract

We show that known Newton-type laws for Optimal Mass Transport, Schrödinger Bridges and the classic Madelung fluid can be derived from variational principles on Wasserstein space. The second order differential equations are accordingly obtained by annihilating the first variation of a suitable action.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    \(\mu _k\) converges weakly to \(\mu \) if \(\int _{{\mathbb R}^N}fd\mu _k\rightarrow \int _{{\mathbb R}^N}fd\mu \) for every continuous, bounded function f.

  2. 2.

    The case of a general reversible Markovian prior is treated in [14] where all the details of the variational analysis may also be found.

References

  1. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, 2nd edn. Birkhäuser Verlag, Basel (2008)

    MATH  Google Scholar 

  2. Ambrosio, L., Gigli, N.: A user’s guide to optimal transport. In: Piccoli, B., Rascle, M. (eds.) Modeling and optimisation of flows on networks. Lecture Notes in Mathematics, vol. 2062, pp. 1–155. Springer, Heidelberg (2013). doi:10.1007/978-3-642-32160-3_1

    Chapter  Google Scholar 

  3. Benamou, J., Brenier, Y.: A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84, 375–393 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Beurling, A.: An automorphism of product measures. Ann. Math. 72, 189–200 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  5. Blaquière, A.: Controllability of a Fokker-Planck equation, the Schrödinger system, and a related stochastic optimal control. Dyn. Control 2(3), 235–253 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. Carlen, E.: Stochastic mechanics: a look back and a look ahead. In: Faris, W.G. (ed.) Diffusion, Quantum Theory and Radically Elementary Mathematics. Mathematical Notes, vol. 47, pp. 117–139. Princeton University Press (2006)

    Google Scholar 

  7. Chen, Y., Georgiou, T.T., Pavon, M.: Optimal steering of a linear stochastic system to a final probability distribution. Part I. IEEE Trans. Aut. Control 61(5), 1158–1169 (2016). arXiv:1408.2222v1

    Article  MATH  MathSciNet  Google Scholar 

  8. Chen, Y., Georgiou, T.T., Pavon, M.: Optimal steering of a linear stochastic system to a final probability distribution. Part II. IEEE Trans. Aut. Control 61(5), 1170–1180 (2016). arXiv:1410.3447v1

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen, Y., Georgiou, T.T., Pavon, M.: Fast cooling for a system of stochastic oscillators. J. Math. Phys. 56(11), 113302 (2015). arXiv:1411.1323v2

    Article  MATH  MathSciNet  Google Scholar 

  10. Chen, Y., Georgiou, T.T., Pavon, M.: On the relation between optimal transport and Schrödinger bridges: a stochastic control viewpoint. J. Optim. Theory Appl. 169(2), 671–691 (2016). doi:10.1007/s10957-015-0803-z. Published Online 2015. arXiv:1412.4430v1

    Article  MATH  MathSciNet  Google Scholar 

  11. Chen, Y., Georgiou, T.T., Pavon, M.: Optimal transport over a linear dynamical system. IEEE Trans. Autom. Control (2017, to appear). arXiv:1502.01265v1

  12. Chen, Y., Georgiou, T.T., Pavon, M.: Entropic and displacement interpolation: a computational approach using the Hilbert metric. SIAM J. Appl. Math. 76(6), 2375–2396 (2016). arXiv:1506.04255v1

    Article  MATH  MathSciNet  Google Scholar 

  13. Conforti, G.: A second order equation for Schrödinger bridges with applications to the hot gas experiment and entropic transportation cost. (2017, preprint). arXiv:1704.04821v2

  14. Conforti, G., Pavon, M.: Extremal flows on Wasserstein space (2017, under preparation)

    Google Scholar 

  15. Cuturi, M.: Sinkhorn distances: lightspeed computation of optimal transport. In: Advances in Neural Information Processing Systems, pp. 2292–2300 (2013)

    Google Scholar 

  16. Dai Pra, P.: A stochastic control approach to reciprocal diffusion processes. Appl. Math. Optim. 23(1), 313–329 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  17. Dai Pra, P., Pavon, M.: On the Markov processes of Schroedinger, the Feynman-Kac formula and stochastic control. In: Kaashoek, M.A., van Schuppen, J.H., Ran, A.C.M. (eds.) Proceedings of MTNS Conference on Realization and Modeling in System Theory, pp. 497–504. Birkaeuser, Boston (1990)

    Google Scholar 

  18. Evans, L.C.: Partial differential equations and Monge-Kantorovich mass transfer. Curr. Dev. Math. 1977, 65–126 (1999). International Press, Boston

    MATH  MathSciNet  Google Scholar 

  19. Fillieger, R., Hongler, M.-O., Streit, L.: Connection between an exactly solvable stochastic optimal control problem and a nonlinear reaction-diffusion equation. J. Optimiz. Theory Appl. 137, 497–505 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Föllmer, H.: Random fields and diffusion processes. In: Hennequin, P.-L. (ed.) École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–1987. LNM, vol. 1362, pp. 101–203. Springer, Heidelberg (1988). doi:10.1007/BFb0086180

    Chapter  Google Scholar 

  21. Fortet, R.: Résolution d’un système d’equations de M. Schrödinger. J. Math. Pure Appl. IX, 83–105 (1940)

    MATH  Google Scholar 

  22. Gangbo, W., Nguyen, T., Tudorascu, A.: Hamilton-Jacobi equations in the Wasserstein space. Methods Appl. Anal. 15(2), 155–183 (2008)

    MATH  MathSciNet  Google Scholar 

  23. Gentil, I., Lonard, C., Ripani, L.: About the analogy between optimal transport and minimal entropy. Ann. Fac. Toulouse (2017, to appear). arXiv: 1510.08230

  24. Guerra, F.: Structural aspects of stochastic mechanics and stochastic field theory. Phys. Rep. 77, 263–312 (1981)

    Article  MathSciNet  Google Scholar 

  25. Guerra, F., Morato, L.: Quantization of dynamical systems and stochastic control theory. Phys. Rev. D 27(8), 1774–1786 (1983)

    Article  MathSciNet  Google Scholar 

  26. Jamison, B.: The Markov processes of Schrödinger. Z. Wahrscheinlichkeitstheorie verw. Gebiete 32, 323–331 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  27. Jiang, X., Luo, Z., Georgiou, T.: Geometric methods for spectral analysis. IEEE Trans. Signal Process. 60(3), 106471074 (2012)

    Article  MathSciNet  Google Scholar 

  28. Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29, 1–17 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  29. Léonard, C.: A survey of the Schroedinger problem and some of its connections with optimal transport. Discrete Contin. Dyn. Syst. A 34(4), 1533–1574 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  30. Léonard, C.: From the Schrödinger problem to the Monge-Kantorovich problem. J. Funct. Anal. 262, 1879–1920 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  31. Madelung, E.: Quantentheorie in hydrodynamischer Form. Z. Phys. 40, 322–326 (1926)

    Article  MATH  Google Scholar 

  32. McCann, R.: A convexity principle for interacting gases. Adv. Math. 128(1), 153–179 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  33. Mikami, T.: Monge’s problem with a quadratic cost by the zero-noise limit of h-path processes. Probab. Theory Relat. Fields 129, 245–260 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  34. Mikami, T., Thieullen, M.: Duality theorem for the stochastic optimal control problem. Stoch. Proc. Appl. 116, 1815–1835 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  35. Mikami, T., Thieullen, M.: Optimal transportation problem by stochastic optimal control. SIAM J. Control Optim. 47(3), 1127–1139 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  36. Nelson, E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton (1967)

    MATH  Google Scholar 

  37. Nelson, E.: Stochastic mechanics and random fields. In: Hennequin, P.-L. (ed.) École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87. LNM, vol. 1362, pp. 427–459. Springer, Heidelberg (1988). doi:10.1007/BFb0086184

    Chapter  Google Scholar 

  38. Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Part. Differ. Equ. 26(1–2), 101–174 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  39. Ollivier, Y., Pajot, H., Villani, C.: Optimal Transportation. Theory and Applications, London Mathematical Society. Lecture Notes Series, vol. 413. Cambridge University Press (2014)

    Google Scholar 

  40. Pavon, M., Wakolbinger, A.: On free energy, stochastic control, and Schroedinger processes. In: Di Masi, G.B., Gombani, A., Kurzhanski, A. (eds.) Modeling, Estimation and Control of Systems with Uncertainty, pp. 334–348. Birkauser, Boston (1991)

    Google Scholar 

  41. Rachev, T., Rüschendorf, L.: Mass Transportation Problems, Volume I: Theory, Volume II: Applications. Probability and Its Applications. Springer, New York (1998)

    Google Scholar 

  42. Sanov, I.S.: On the probability of large deviations of random magnitudes (in Russian). Mat. Sb. N. S. 42(84) 111744 (1957). Select. Transl. Math. Stat. Probab. 1, 213–244 (1961)

    Google Scholar 

  43. Santambrogio, F.: Optimal Transport for Applied Mathematicians. Birkhäuser (2015)

    Google Scholar 

  44. Schrödinger, E.: Über die Umkehrung der Naturgesetze. Sitzungsberichte der Preuss Akad. Wissen. Berlin Phys. Math. Klasse, 144–153 (1931)

    Google Scholar 

  45. Schrödinger, E.: Sur la théorie relativiste de l’electron et l’interpretation de la mécanique quantique. Ann. Inst. H. Poincaré 2, 269 (1932)

    MATH  MathSciNet  Google Scholar 

  46. Villani, C.: Topics in Optimal Transportation, vol. 58. AMS (2003)

    Google Scholar 

  47. Villani, C.: Optimal Transport. Old and New. Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer, Heidelberg (2009)

    Book  MATH  Google Scholar 

  48. von Renesse, M.-K.: An optimal transport view of Schrödinger’s equation. Canad. Math. Bull. 55, 858–869 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  49. Yasue, K.: Stochastic calculus of variations. J. Funct. Anal. 41, 327–340 (1981)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire Paul Painlevé, Université des Sciences et Technologies de Lille 1, 59655, Villeneuve d’Ascq Cedex, France

    Giovanni Conforti

  2. Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova, via Trieste 63, 35121, Padova, Italy

    Michele Pavon

Authors
  1. Giovanni Conforti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michele Pavon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michele Pavon .

Editor information

Editors and Affiliations

  1. Ecole Polytechnique, Palaiseau, France

    Frank Nielsen

  2. Thales Land and Air Systems, Limours, France

    Frédéric Barbaresco

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Conforti, G., Pavon, M. (2017). Extremal Curves in Wasserstein Space. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science(), vol 10589. Springer, Cham. https://doi.org/10.1007/978-3-319-68445-1_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-68445-1_11

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-68444-4

  • Online ISBN: 978-3-319-68445-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics