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Hopf–Cole Transformation and Schrödinger Problems

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Geometric Science of Information (GSI 2019)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 11712))

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Abstract

We study generalized Hopf–Cole transformations motivated by the Schrödinger bridge problem. We present two examples of canonical transformations, including a Schrödinger problem associated with a quadratic Rényi entropy.

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References

  1. Ambrosio, L., Gangbo, W.: Hamiltonian ODEs in the Wasserstein space of probability measures. Commun. Pure and Appl. Math.: A J. Issued Courant Inst. Math. Sci. 61(1), 18–53 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Beurling, A.: An automorphism of product measures. Ann. Math. 72(1), 189–200 (1960)

    Article  MathSciNet  Google Scholar 

  4. Carlen, E.A.: Stochastic Mechanics: A Look Back and a Look Ahead. Princeton University Press, Berlin (2014)

    Google Scholar 

  5. Chow, S.-N., Li, W., Zhou, H.: A discrete Schrödinger equation via optimal transport on graphs. J. Funct. Anal. 276(8), 2440–2469 (2019)

    Article  MathSciNet  Google Scholar 

  6. Conforti, G.: A second order equation for Schrödinger bridges withapplications to the hot gas experiment and entropic transportation cost. Probab. Theor. Relat. Fields 174, 1–47 (2018)

    Article  Google Scholar 

  7. Conforti, G., Pavon, M.: Extremal flows in Wasserstein space. J. Math. Phys. 59(6), 063502 (2018)

    Article  MathSciNet  Google Scholar 

  8. Föllmer, H.: Random fields and diffusion processes. In: Hennequin, P.-L. (ed.) École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87. LNM, vol. 1362, pp. 101–203. Springer, Heidelberg (1988). https://doi.org/10.1007/BFb0086180

    Chapter  MATH  Google Scholar 

  9. Fortet, R.: Résolution d’un système d’équations de M. Schrödinger. J. Math. Pures Appl. 19, 83–105 (1940)

    MathSciNet  MATH  Google Scholar 

  10. Gentil, I., Léonard, C., Ripani, L.: Dynamical aspects of generalized Schrödinger problem via Otto calculus – a heuristic point of view. arXiv:1806.01553 (2018)

  11. Jamison, B.: The Markov processes of Schrödinger. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 32(4), 323–331 (1975)

    Article  Google Scholar 

  12. Khesin, B., Misiolek, G., Modin, K.: Geometric hydrodynamics via Madelung transform. Proc. Nat. Acad. Sci. 115(24), 6165–6170 (2018)

    Article  MathSciNet  Google Scholar 

  13. Lafferty, J.D.: The density manifold and configuration space quantization. Trans. Am. Math. Soc. 305(2), 699–699 (1988)

    Article  MathSciNet  Google Scholar 

  14. Léger, F.: A geometric perspective on regularized optimal transport. J. Dyn. Differ. Equ. 2018, 1–15 (2018)

    Google Scholar 

  15. Léger, F., Li, W.: Hopf-Cole transformation via generalized Schrödinger bridge problem. arXiv:1901.09051 (2019)

  16. Léonard, C.: A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete and Continuous Dyn. Syst. 34(4), 1533–1574 (2013)

    Article  Google Scholar 

  17. Li, W.: Geometry of probability simplex via optimal transport. arXiv:1803.06360 (2018)

  18. Nelson, E.: Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev. 150(4), 1079–1085 (1966)

    Article  Google Scholar 

  19. Nelson, E.: Quantum Fluctuations. Princeton Series in Physics. Princeton University Press, Princeton (1985)

    MATH  Google Scholar 

  20. Schrödinger, E.: Über die Umkehrung der Naturgesetze. Sitzungsber. Preuß. Akad. Wiss. Phys. Math.- Kl 9, 144–153 (1931)

    MATH  Google Scholar 

  21. Schrödinger, E.: Sur la théorie relativiste de l’électron et l’interprétation de la mécanique quantique. Ann. Inst. Henri Poincaré 2(4), 269–310 (1932)

    MathSciNet  MATH  Google Scholar 

  22. Villani, C.: Topics in Optimal Transportation, Graduate studies in mathematics, vol. 58. American Mathematical Society, Providence (2003)

    MATH  Google Scholar 

  23. von Renesse, M.-K.: An optimal transport view of Schrödinger’s equation. Canad. Math. Bull. 55(4), 858–869 (2012)

    Article  MathSciNet  Google Scholar 

  24. Yasue, K.: Stochastic calculus of variations. J. Funct. Anal. 41(3), 327–340 (1981)

    Article  MathSciNet  Google Scholar 

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Acknowledgement

This project has received funding from AFOSR MURI FA9550-18-1-0502.

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Correspondence to Flavien Léger .

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Léger, F., Li, W. (2019). Hopf–Cole Transformation and Schrödinger Problems. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science(), vol 11712. Springer, Cham. https://doi.org/10.1007/978-3-030-26980-7_76

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  • DOI: https://doi.org/10.1007/978-3-030-26980-7_76

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26979-1

  • Online ISBN: 978-3-030-26980-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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