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A second order equation for Schrödinger bridges with applications to the hot gas experiment and entropic transportation cost

  • Giovanni ConfortiEmail author
Article

Abstract

The Schrödinger problem is obtained by replacing the mean square distance with the relative entropy in the Monge–Kantorovich problem. It was first addressed by Schrödinger as the problem of describing the most likely evolution of a large number of Brownian particles conditioned to reach an “unexpected configuration”. Its optimal value, the entropic transportation cost, and its optimal solution, the Schrödinger bridge, stand as the natural probabilistic counterparts to the transportation cost and displacement interpolation. Moreover, they provide a natural way of lifting from the point to the measure setting the concept of Brownian bridge. In this article, we prove that the Schrödinger bridge solves a second order equation in the Riemannian structure of optimal transport. Roughly speaking, the equation says that its acceleration is the gradient of the Fisher information. Using this result, we obtain a fine quantitative description of the dynamics, and a new functional inequality for the entropic transportation cost, that generalize Talagrand’s transportation inequality. Finally, we study the convexity of the Fisher information along Schrödigner bridges, under the hypothesis that the associated reciprocal characteristic is convex. The techniques developed in this article are also well suited to study the Feynman–Kac penalisations of Brownian motion.

Mathematics Subject Classification

60J60 39B62 60F10 46N10 47D07 

Notes

Acknowledgements

The author acknowledges support from CEMPI Lille and the University of Lille 1. He also wishes to thank Christian Léonard for having introduced him to the Schrödinger problem, and for many fruitful discussions.

References

  1. 1.
    Ambrosio, L., Gangbo, W.: Hamiltonian odes in the wasserstein space of probability measures. Commun. Pure Appl. Math. 61(1), 18–53 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Ambrosio, L., Gigli, N.: A user’s guide to optimal transport. In: Piccoli, B., Rascle, M. (eds.) Modelling and Optimisation of Flows on Networks: Cetraro, Italy 2009, pp. 1–155. Springer, Berlin, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-32160-3_1 CrossRefGoogle Scholar
  3. 3.
    Bakry, D., Gentil, I., Ledoux, M.: Analysis and Geometry of Markov Diffusion Operators, vol. 348. Springer, Berlin (2013)zbMATHGoogle Scholar
  4. 4.
    Benamou, J.-D., Brenier, Y.: A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84(3), 375–393 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Benamou, J.-D., Carlier, G., Cuturi, M., Nenna, L., Peyré, G.: Iterative bregman projections for regularized transportation problems. SIAM J. Sci. Comput. 37(2), A1111–A1138 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cattiaux, P., Léonard, C.: Minimization of the Kullback information of diffusion processes. Annal. de l’IHP Probab. et Stat. 30(1), 83–132 (1994)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chen, Y., Tryphon, T.G., Michele, P.: On the relation between optimal transport and Schrödinger bridges: a stochastic control viewpoint. J. Optim. Theory Appl. 169(2), 671–691 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chen, Y., Tryphon, T.G.: Optimal steering of a linear stochastic system to a final probability distribution, part i. IEEE Trans. Autom. Control 61(5), 1158–1169 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Chen, Y., Tryphon, T.G.: Optimal steering of a linear stochastic system to a final probability distribution, part ii. IEEE Trans. Autom. Control 61(5), 1170–1180 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Chow, S.-N., Li, W., Zhou, H.: A discrete schrodinger equation via optimal transport on graphs. arXiv preprint arXiv:1705.07583 (2017)
  11. 11.
    Clark, J.M.C.: A local characterization of reciprocal diffusions. Appl. Stoch. Anal. 5, 45–59 (1991)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Conforti, G.: Fluctuations of bridges, reciprocal characteristics, and concentration of measure. preprint arXiv:1602.07231 to appear in Annales de l’Institut Henri Poincaré (2016)
  13. 13.
    Conforti, G., Léonard, C.: Reciprocal classes of random walks on graphs. Stoch. Process. Appl. 127(6), 1870–1896 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Conforti, G., Von Renesse, M.: Couplings, gradient estimates and logarithmic Sobolev inequality for Langevin bridges. Probab. Theory Related Fields (2017). available onlineGoogle Scholar
  15. 15.
    Cruzeiro, A.B., Zambrini, J.C.: Malliavin calculus and Euclidean quantum mechanics. I. Functional calculus. J. Funct. Anal. 96(1), 62–95 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Dai Pra, P.: stochastic control approach to reciprocal diffusion processes. Appl. Math. Optim. 23(1), 313–329 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Dawson, D., Gorostiza, L., Wakolbinger, A.: Schrödinger processes and large deviations. J. Math. Phys. 31(10), 2385–2388 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Donald, A.: Dawson and Jürgen Gärtner. Multilevel large deviations and interacting diffusions. Probab. Theory Relat. Fields 98(4), 423–487 (1994)CrossRefGoogle Scholar
  19. 19.
    Do Carmo, M.P., Flaherty, J.F.: Riemannian Geometry, vol. 115. Birkhäuser, Boston (1992)CrossRefGoogle Scholar
  20. 20.
    Föllmer, H.: Random fields and diffusion processes. In École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87, pp. 101–203. Springer (1988)Google Scholar
  21. 21.
    Föllmer, H., Gantert, N., et al.: Entropy minimization and schrödinger processes in infinite dimensions. Ann. Probab. 25(2), 901–926 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Galichon, A., Kominers, S.D., Weber, S.: The nonlinear bernstein-schrödinger equation in economics. In: Nielsen, F., Barbaresco, F. (eds.) Geometric Science of Information, pp. 51–59. Springer International Publishing, Cham (2015)CrossRefGoogle Scholar
  23. 23.
    Gentil, I., Léonard, C., Ripani, L.: About the analogy between optimal transport and minimal entropy. Annales de la facultés des sciences de Toulouse Sér. 6 26(3), 569–700 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Gianazza, U., Savaré, G., Toscani, G.: The wasserstein gradient flow of the fisher information and the quantum drift-diffusion equation. Arch. Ration. Mech. Anal. 194(1), 133–220 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Gigli, N.: Second Order Analysis on (P2(M),W2). Memoirs of the American Mathematical Society, Providence (2012)Google Scholar
  26. 26.
    Gigli, N., Tamanini, L.: Second order differentiation formula on compact RCD*(K, N) spaces. arXiv preprint arXiv:1701.03932 (2017)
  27. 27.
    Gozlan, N., Roberto, C., Samson, P.-M., Tetali, P.: Kantorovich duality for general transport costs and applications. J. Funct. Anal. 273(11), 3327–3405 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge university press, Cambridge (2012)CrossRefGoogle Scholar
  29. 29.
    Krener, A.J.: Reciprocal diffusions and stochastic differential equations of second order. Stochastics 107(4), 393–422 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Krener, A.J.: Reciprocal diffusions in flat space. Probab. Theory Relat. Fields 107(2), 243–281 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Léonard, C.: From the Schrödinger problem to the Monge–Kantorovich problem. J. Funct. Anal. 262(4), 1879–1920 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Léonard, C.: A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete Contin. Dyn. Syst. 34(4), 1533–1574 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Léonard, C., Rœlly, S., Zambrini, J.C.: Reciprocal processes. A measure-theoretical point of view. Probab. Surv. 11, 237–269 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Léonard, C.: Some properties of path measures. In Séminaire de Probabilités XLVI, pp. 207–230. Springer (2014)Google Scholar
  35. 35.
    Léonard, C.: On the convexity of the entropy along entropic interpolations. In: Gigli, N. (ed.) Measure Theory in Non-Smooth Spaces, Partial Differential Equations and Measure Theory. De Gruyter Open, Berlin (2017)Google Scholar
  36. 36.
    Léonard, C., et al.: Lazy random walks and optimal transport on graphs. Ann. Probab. 44(3), 1864–1915 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Levy, B.C., Krener, A.J.: Dynamics and kinematics of reciprocal diffusions. J. Math. Phys. 34(5), 1846–1875 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Li, W., Yin, P., Osher, S.: Computations of optimal transport distance with fisher information regularization. J. Sci. Comput. 19, 1–15 (2017)zbMATHCrossRefGoogle Scholar
  39. 39.
    Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169, 903–991 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Mikami, T.: Monge’s problem with a quadratic cost by the zero-noise limit of h-path processes. Probab. Theory Relat. Fields 129(2), 245–260 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Nelson, E.: Dynamical Theories of Brownian Motion, vol. 2. Princeton University Press, Princeton (1967)zbMATHGoogle Scholar
  42. 42.
    Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Otto, F., Villani, C.: Generalization of an inequality by talagrand and links with the logarithmic sobolev inequality. J. Funct. Anal. 173(2), 361–400 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Rœlly, S., Thieullen, M.: A characterization of reciprocal processes via an integration by parts formula on the path space. Probab. Theory Relat. Fields 123(1), 97–120 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Rœlly, S., Thieullen, M.: Duality formula for the bridges of a brownian diffusion: application to gradient drifts. Stoch. Process. Appl. 115(10), 1677–1700 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Roynette, B., Yor, M.: Penalising Brownian Paths, vol. 1969. Springer, Berlin (2009)zbMATHCrossRefGoogle Scholar
  47. 47.
    Rüschendorf, L., Thomsen, W.: Note on the Schrödinger equation and I-projections. Stat. Probab. Lett. 17(5), 369–375 (1993)zbMATHCrossRefGoogle Scholar
  48. 48.
    Schrödinger, E.: Über die Umkehrung der Naturgesetze. Sitzungsberichte Preuss. Akad. Wiss. Berlin. Phys. Math. 144, 144–153 (1931)zbMATHGoogle Scholar
  49. 49.
    Schrödinger, E.: La théorie relativiste de l’électron et l’ interprétation de la mécanique quantique. Ann. Inst Henri Poincaré 2, 269–310 (1932)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Solomon, J., De Goes, F., Peyré, G., Cuturi, M., Butscher, A., Nguyen, A., Tao, D., Guibas, L.: Convolutional wasserstein distances: Efficient optimal transportation on geometric domains. ACM Trans. Grap.(TOG) 34(4), 66 (2015)zbMATHGoogle Scholar
  51. 51.
    Sturm, K.-T.: On the geometry of metric measure spaces. Acta Math. 196(1), 65–131 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Talagrand, M.: Transportation cost for gaussian and other product measures. Geom. Funct. Anal. GAFA 6(3), 587–600 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Thieullen, M.: Second order stochastic differential equations and non-Gaussian reciprocal diffusions. Probab. Theory Relat. Fields 97(1–2), 231–257 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Villani, C.: Optimal Transport: Old and New, vol. 338. Springer, Berlin (2008)zbMATHGoogle Scholar
  55. 55.
    von Renesse, M.-K.: An optimal transport view of Schrödinger’s equation. Can. Math. Bull. 55(4), 858–869 (2012)zbMATHCrossRefGoogle Scholar
  56. 56.
    von Renesse, M.-K., Sturm, K.-T.: Transport inequalities, gradient estimates, entropy and Ricci curvature. Commun. Pure Appl. Math. 58(7), 923–940 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Wakolbinger, A.: A simplified variational characterization of Schrödinger processes. J. Math. Phys. 30(12), 2943–2946 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Zambrini, J.C.: Variational processes and stochastic versions of mechanics. J. Math. Phys. 27(9), 2307–2330 (1986)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Département de Mathématiques AppliquéesÉcole PolytechniquePalaiseau CedexFrance

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