Some Results on Entropic Projections

  • C. Léonard
Part of the Progress in Probability book series (PRPR, volume 50)


We give a short survey of results related to the maximum entropy method. In this article we use the large deviations approach rather than the more direct convex analytical one. Indeed, the proposed applications are naturally stated in terms of large random particle systems, namely, the existence and construction problems for Schrödinger’s bridges and Nelson’s diffusion processes. These problems arise from probabilistic approaches to quantum mechanics.


Relative Entropy Maximum Entropy Method Large Deviation Principle Contraction Principle Effective Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Maximum Entropy and Bayesian Methods, Proceedings of the 11th International Workshop on Maximum Entropy and Bayesian Methods of Statistical Analysis, eds. C. R. Smith, G. J. Erickson and P. O. Neudorfer, Seattle, Kluwer, 1991.Google Scholar
  2. [2]
    S. Bernstein, Sur les liaisons entre les grandeurs aléatoires Vehr. des intern. Mathematikerkongr., Zürich 1932, Band 1.Google Scholar
  3. [3]
    A. Beurling, An automorphism of product measures, Ann. Math., 72 (1960), 189–200.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    J. M. Borwein and A. S. Lewis, Partially-finite programming in L1 and the existence of maximum entropy estimates SIAM J. Optim., 3 (1993), 248–267.MathSciNetMATHGoogle Scholar
  5. [5]
    P. Cattiaux and C. Léonard, Minimization of the Kullback information of diffusion processes, Ann. Inst. Henri Poincaré, 30 (1994), 83–132.MATHGoogle Scholar
  6. [6]
    P. Cattiaux and C. Léonard, Large deviations and Nelson processes, Forum Math., 7 (1995), 95–115.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    E. Carlen, Conservative diffusions, Comm. Math. Phys., 94 (1984), 293–315.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    A.B. Cruzeiro, L. Wu, J.C. Zambrini, Bernstein processes associated with a Markov process. Stochastic Analysis and Mathematical Physics, (Santiago 98, Ed. R. Rebolledo). Trends in Math., Birkhaüser, 2000, 41–72.CrossRefGoogle Scholar
  9. [9]
    A.B.Cruzeiro, J.C. Zambrini, Malliavin calculus and euclidean quantum mechanics, I., J. Funct. Anal, 96:1 (1991), 62–95.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    I. Csiszár, I-Divergence Geometry of Probability Distributions and Minimization Problems, Ann. Probab., 3(1975), 146–158.MATHCrossRefGoogle Scholar
  11. [11]
    I. Csiszár, Sanov property, generalized I-projection and a conditional limit theorem, Ann. Probab., 12(1984), 768–793.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    I. Csiszár, F. Gamboa and E. Gassiat, MEM pixel correlated solutions for generalized moment and interpolation problems, IEEE Trans. Inform. Theory., 45(Nov. 1999), 2253–2270.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Second edition, Springer-Verlag, 1998.MATHCrossRefGoogle Scholar
  14. [14]
    P. Eischelsbacher and U. Schmock, Large and moderate deviations of products of empirical measures aud U-empirical processes in strong topologies, preprint, 1997.Google Scholar
  15. [15]
    P. Eischelsbacher and U. Schmock, Exponential approximations in completely regular topological spaces and extensions of Sanov’s theorem, Stochastic Processes and their Applications, 11 (1998), 233–251.CrossRefGoogle Scholar
  16. [16]
    H. Föllmer, Random Fields and Diffusion Processes, Cours à l’Ecole d’Été de Probabilités de Saint-Flour, Lecture Notes in Math. 1362, Springer Verlag, 1988.Google Scholar
  17. [17]
    R. Fortet, Résolution d’un système d’équations de M. Schrödinger, J. Math. Pures. Appl., IX(1940), 83–105.MathSciNetGoogle Scholar
  18. [18]
    C. Léonard, Minimizers of energy functionals, to appear in Acta Mathematica Hungarica.Google Scholar
  19. [19]
    C. Léonard, Minimization of energy functionals applied to some inverse problems, to appear in Journal of Applied Mathematics and Optimization.Google Scholar
  20. [20]
    C. Léonard, Minimizers of energy functionals under not very integrable constraints, preprint Ecole Polytechnique, CMAP, 2000.Google Scholar
  21. [21]
    C. Léonard and J. Najim, An extension of Sanov’s theorem. Application to the Gibbs conditioning principle, preprint Ecole Polytechnique, CMAP, 2000.Google Scholar
  22. [22]
    F. Liese and I. Vajda, Convex Statistical Distances, Leipzig, B.G. Teubner, 1987.Google Scholar
  23. [23]
    E. Nelson, Quantum Fluctuations, Princeton Series in Physics, Princeton University Press, 1985.Google Scholar
  24. [24]
    L. Rüschendorf, On the minimum discrimination information theorem, Statist. Decisions, Supplementary volume 1, 1984.Google Scholar
  25. [25]
    A. Schied, Cramér’s condition and Sanov’s theorem, Statitics and Probability Letters, 39(1998), 55–60.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    E. Schrödinger, Sur la théorie relativiste de l’électron et l’interprétation de la mécanique quantique. Ann. Inst. Henri Poincaré, 2(1932), 269–310.Google Scholar
  27. [27]
    D.W. Stroock and O. Zeitouni, Microcanonical distributions, Gibbs states, and the equivalence of ensembles, in: Festchrift in honour of F. Spitzer, R. Durrett and H. Kesten, eds., Springer Science+Business Media New York, 1991, 399–424.Google Scholar
  28. [28]
    M. Teboulle and I. Vajda, Convergence of Best ø-Entropy Estimates, IEEE Trans. Inform. Theory, 9(Jan. 1993), 297–301.MathSciNetGoogle Scholar
  29. [29]
    L. Wu, Uniqueness of Nelson’s diffusions, Probab. Th. and Rel. Fields, 114 (1999), 549–585.MATHCrossRefGoogle Scholar
  30. [30]
    J.C. Zambrini, Variational processes and stochastic versions of mechanics, J. Math. Phys., 9(1986), 2307–2330.MathSciNetCrossRefGoogle Scholar
  31. [31]
    W.A. Zheng, Tightness results for laws of diffusion processes, application to stochastic mechanics, Ann. Inst. Henri Poincaré, B 21(1985), 103–124.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • C. Léonard
    • 1
    • 2
  1. 1.Modal-X, Université Paris 10, Bât. GNanterre CedexFrance
  2. 2.Centre de Mathématiques Appliquées, Ecole PolytechniquePalaiseau CedexFrance

Personalised recommendations