Local Invertibility of Adapted Shifts on Wiener Space and Related Topics

  • Rémi LassalleEmail author
  • A. S. Üstünel
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 34)


In this article we show that the invertibility of an adapted shift on the Brownian sheet is a local property in the usual sense of stochastic calculus. Thanks to this result we give a short proof of the invertibility for some processes which occur in free euclidean quantum mechanics and we relate this result to optimal transport. We also investigate some applications to information theory of a recent criterion which relates the invertibility of a shift to an equality between the energy of the signal and the relative entropy of the measure it induces. In particular, thanks to a change of measure, we interpret Shannon’s inequality as a consequence of information loss in Gaussian channels and we extend it to any abstract Wiener space. Finally, we extend the criterion of invertibility to the case of some stochastic differential equations with dispersion.


Strong solutions Entropy Invertibility Shannon’s inequality 


  1. 1.
    Artstein, S., Ball, K.M., Barthe, F., Naor, A.: Solution of Shannon’s problem on the monotonicity of entropy. J. Amer. Math. Soc. 17(4), 975–982 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Carlen, E., Cordero-Erausquin, D.: Subadditivity of the entropy and its relation to Brascamp–Lieb type inequalities. Geom. Funct. Anal. 19, 373–405 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Dellacherie, C., Meyer, P.A.: Probabilités et Potentiel Ch. I à IV. Hermann, Paris (1975)Google Scholar
  4. 4.
    Dellacherie, C., Meyer, P.A.: Probabilités et Potentiel Ch. V à VIII. Hermann, Paris (1980)Google Scholar
  5. 5.
    Feyel, D., Üstünel, A.S.: Monge-Kantorovitch measure transportation and Monge-Ampère equation on Wiener space. Probab. Theor. Relat. Fields 128(3), 347–385 (2004)zbMATHCrossRefGoogle Scholar
  6. 6.
    Feyel, D., Üstünel, A.S.: Solution of the Monge-Ampère equation on Wiener space for general log-concave measures. J. Funct. Anal. 232(1), 29–55 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Föllmer, H.: Time reversal on Wiener space. In: Lecture Notes in Mathematics. Springer, Berlin (1986)Google Scholar
  8. 8.
    Föllmer, H.: Time reversal of infinite-dimensional diffusions. Stoch. Proc. Appl. 22(1), 59–77 (1986)zbMATHCrossRefGoogle Scholar
  9. 9.
    Föllmer, H.: Random fields and diffusion processes. Ecole d’Été de Saint Flour XV-XVII. In: Lecture Notes in Mathematics, vol. 1362, pp. 101–203. Springer, New York (1988)Google Scholar
  10. 10.
    Föllmer, H., Gandert, N.: Entropy minimization and Schrödinger process in infinite demensions. Ann. Probab. 25(2), 901–926 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Gross, L.: Logarithmic Sobolev inequalities. Amer. J. Math. 97, 1061–1083 (1975)Google Scholar
  12. 12.
    Guerra, F., Morato, L.: Quantization of dynamical systems and stochastic control theory. Phys. Rev. D 27(8) 1774–1786 (1983)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North Holland, Amsterdam (Kodansha Ltd., Tokyo) (1981)Google Scholar
  14. 14.
    Krylov, N.V., Röckner, M.R.: Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Relat. Fields 131(2), 154–196 (2005)zbMATHCrossRefGoogle Scholar
  15. 15.
    Kuo, H.: Gaussian measures in Banach spaces. In: Lecture Notes in Mathematics, vol. 463. Springer, Berlin (1975)Google Scholar
  16. 16.
    Lassalle, R.: Invertibility of adapted perturbations of the identity on abstract Wiener space. J. Func. Anal. 262(6), 2734–2776 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Lassalle, R.: Local invertibility of adapted shifts on Wiener space, under finite energy condition, Stochastics, An International Journal of Probability and Stochastics Processes, (2012) DOI: 10.1080/17442508.2012.720257Google Scholar
  18. 18.
    Lehec, J.: A stochastic formula for the entropy and applications. (2010) (Preprint)Google Scholar
  19. 19.
    Malliavin, P.: Stochastic Analysis. Springer, New York (1997)zbMATHGoogle Scholar
  20. 20.
    Mikami, T.: Dynamical systems in the variational formulation of the Fokker-Planck equation by the Wasserstein metric, Series 464 (1999). Available at
  21. 21.
    Mikami, T.: Optimal control for absolutely continuous stochastic processes and the mass transportation problem. Elect. Comm. Probab. 7, 199–213 (2002)MathSciNetGoogle Scholar
  22. 22.
    Mikami, T., Thieullen, M.: Duality theorem for stochastic optimal control problem. Stoch. Proc. Appl. 116, 1815–1835 (2006). MR230760Google Scholar
  23. 23.
    Nelson, E.: Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev. 150(4) (1966)Google Scholar
  24. 24.
    Nelson, E.: Quantum Fluctuations. Princeton Series in Physics. Princeton University Press, NJ (1985)zbMATHGoogle Scholar
  25. 25.
    Nelson, E.: Dynamical Theories of Brownian Motion. Princeton university Press, NJ (1967)zbMATHGoogle Scholar
  26. 26.
    Nualart, D.: The Malliavin calculus and related topics. Probability and its Applications, vol. 21. Springer, Berlin (1995)Google Scholar
  27. 27.
    Talagrand, M.: Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6, 587–600 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Üstünel, A.S.: Introduction to analysis on Wiener space. In: Lecture Notes in Mathematics, vol. 1610. Springer, Berlin (1995)Google Scholar
  29. 29.
    Üstünel, A.S.: A necessary and sufficient condition for the invertibility of adapted perturbations of identity on the Wiener space. C.R. Acad. Sci. Paris, Ser. I 346, 97–900 (2008)Google Scholar
  30. 30.
    Üstünel, A.S.: Entropy, invertibility and variational calculus of adapted shifts on Wiener space. J. Funct. Anal. 257(11), 3655–3689 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Üstünel, A.S.: Analysis on Wiener Space and Applications (2010) arxivGoogle Scholar
  32. 32.
    Üstünel, A.S.: Persistence of invertibility on the Wiener space. Commun. Stoch. Anal. 4(2), 201–213 (2010)MathSciNetGoogle Scholar
  33. 33.
    Üstünel, A.S., Zakai, M.: The construction of filtrations on abstract Wiener space. J. Funct. Anal. 143, 10–32 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Üstünel, A.S., Zakai, M.: Transformation of Measure on Wiener Space. Springer, Berlin (1999)Google Scholar
  35. 35.
    Üstünel, A.S., Zakai, M.: Sufficient conditions for the invertibility of adapted perturbations of identity on the Wiener space. Probab. Theory Relat. Fields 139, 207–234 (2007)zbMATHCrossRefGoogle Scholar
  36. 36.
    Veretennikov, A.Yu.: On strong solutions and explicit formulas for solutions of stochastic integral equations. Mat. Sb. (N.S.) 111(153)(3), 434–452 (1980)Google Scholar
  37. 37.
    Villani, C.: Topics in optimal transportation. Graduate Series in Mathematics, vol. 58. American Mathematical Society, Providence (2003)Google Scholar
  38. 38.
    Yasue, K.: Stochastic calculus of variations. J. Funct. Anal. (1981)Google Scholar
  39. 39.
    Zambrini, J.C.: Stochastic mechanics according to E. Schrödinger. Phys. Rev. A 33(3) (1986)Google Scholar
  40. 40.
    Zambrini, J.C.: Variational processes and stochastic versions of mechanics. J. math. Phys. 27 (9), 2307–2330 (1986)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.LTCI CNRS Dépt. Infres, Institut Telecom, Telecom ParisTechParisFrance

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