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Monge’s problem with a quadratic cost by the zero-noise limit of h-path processes
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  • Published: 25 March 2004

Monge’s problem with a quadratic cost by the zero-noise limit of h-path processes

  • Toshio Mikami1 

Probability Theory and Related Fields volume 129, pages 245–260 (2004)Cite this article

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Abstract.

We study the asymptotic behavior, in the zero-noise limit, of solutions to Schrödinger’s functional equations and that of h-path processes, and give a new proof of the existence of the minimizer of Monge’s problem with a quadratic cost.

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References

  1. Ambrosio, L., Pratelli, A.: Existence and stability results in the L 1 theory of optimal transportation. Preprint

  2. Bakelman, I.J.: Convex Analysis and Nonlinear Geometric Elliptic Equations. (Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1994)

  3. Brenier, Y.: Décomposition polaire et réarrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris Sć4r. I Math. 305, 805–808 (1987)

  4. Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44, 375–417 (1991)

    Google Scholar 

  5. Brenier, Y., Benamou, J.D.: A numerical method for the optimal mass transport problem and related problems. In: Monge Ampre equation: applications to geometry and optimization, Proceedings of the NSF-CBMS Conference, Deerfield Beach, FL, 1997 (Caffarelli, L.A., Milman, M., (ed.)), Contemporary Mathematics 226, (Amer. Math. Soc., Providence, RI, 1999) pp. 1–11

  6. Caffarelli, L.A., Feldman, M., McCann, R.J.: Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs. J. of the Amer. Math. Soc. 15, 1–26 (2001)

    Google Scholar 

  7. Doob, J.L.: Classical potential theory and its probabilistic counterpart. (Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1984)

  8. Evans, L,C,: Partial differential equations and Monge-Kantorovich mass transfer. Current developments in mathematics, 1997 (Cambridge, MA) (Int. Press, Boston, MA, 1999) pp. 65–126

  9. Evans, L.C., Gangbo, W.: Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc. 137 (653), (1999)

  10. Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. (Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1993)

  11. Föllmer, H.: Random fields and diffusion processes. In: École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87 (Hennequin, P.L., (ed.)), Lecture Notes in Math. 1362 (Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1988) pp. 101–203

  12. Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. (North-Holland/Kodansha, Amsterdam, New York, Oxford, Tokyo, 1981)

  13. Jamison, B.: Reciprocal processes. Z. Wahrsch. Verw. Gebiete 30, 65–86 (1974)

    Google Scholar 

  14. Jamison, B.: The Markov process of Schrödinger: Z. Wahrsch. Verw. Gebiete 32, 323–331 (1975)

    Google Scholar 

  15. Knott., M., Smith, C.S.: On the optimal mapping of distributions. J. of Optimization Theory and Applications 43, 39–49 (1984)

    Google Scholar 

  16. Knott, M., Smith, C.S.: Note on the optimal transportation of distributions. J. of Optimization Theory and Applications 52, 323–329 (1987)

    Google Scholar 

  17. Mikami, T.: Variational processes from the weak forward equation. Commun. Math. Phys. 135, 19–40 (1990)

    Google Scholar 

  18. Mikami, T.: Dynamical systems in the variational formulation of the Fokker-Planck equation by the Wasserstein metric. Appl. Math. Optim. 42, 203–227 (2000)

    Google Scholar 

  19. Mikami, T.: Optimal control for absolutely continuous stochastic processes and the mass transportation problem. Elect. Comm. in Probab. 7, 199–213 (2002)

    Google Scholar 

  20. Rachev, S.T., Rüschendorf, L.: Mass transportation problems, Vol. I: Theory. (Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1999)

  21. Rüschendorf, L., Rachev, S.T.: A characterization of random variables with minimum L 2-distance. J. Multivarite Anal. 32, 48–54 (1990)

    Google Scholar 

  22. Rüschendorf, L., Thomsen, W.: Note on the Schrödinger equation and I-projections. Statist. Probab. Lett. 17, 369–375 (1993)

    Google Scholar 

  23. Salisbury, T.S.: An increasing diffusion. In: Seminar on Stochastic Processes 1984 (Cinlar, E., Chung, K.L., Getoor, R.K., (eds.)) (Birkhäuser, Boston, MA, 1986) pp. 173–194

  24. Trudinger, N.S., Xu-Jia, W.: On the Monge mass transfer problem. Calc. Var. 13, 19–31 (2001)

    Google Scholar 

  25. Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics Vol. 58, (Providence, RI: Amer. Math. Soc., 2003)

  26. Zambrini, J.C.: Variational processes. In: Stochastic processes in classical and quantum systems, Ascona, 1985, (Albeverio, S., Casati, G., Merlini, D., (eds.)), Lecture Notes in Phys. 262, (Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1986) 517–529.

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Authors and Affiliations

  1. Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan

    Toshio Mikami

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  1. Toshio Mikami
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Correspondence to Toshio Mikami.

Additional information

Partially supported by the Grant-in-Aid for Scientific Research, No. 15340047 and 15340051, JSPS.

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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). https://doi.org/10.1007/s00440-004-0340-4

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  • Received: 10 August 2003

  • Revised: 10 December 2003

  • Published: 25 March 2004

  • Issue Date: June 2004

  • DOI: https://doi.org/10.1007/s00440-004-0340-4

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Keywords

  • Asymptotic Behavior
  • Functional Equation
  • Quadratic Cost
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