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Large deviations for many Brownian bridges with symmetrised initial-terminal condition
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  • Published: 06 September 2007

Large deviations for many Brownian bridges with symmetrised initial-terminal condition

  • Stefan Adams1 &
  • Wolfgang König2 

Probability Theory and Related Fields volume 142, pages 79–124 (2008)Cite this article

Abstract

Consider a large system of N Brownian motions in \({\mathbb{R}}^d\) with some non-degenerate initial measure on some fixed time interval [0,β] with symmetrised initial-terminal condition. That is, for any i, the terminal location of the i-th motion is affixed to the initial point of the σ(i)-th motion, where σ is a uniformly distributed random permutation of 1,...,N. Such systems play an important role in quantum physics in the description of Boson systems at positive temperature 1/β. In this paper, we describe the large-N behaviour of the empirical path measure (the mean of the Dirac measures in the N paths) and of the mean of the normalised occupation measures of the N motions in terms of large deviations principles. The rate functions are given as variational formulas involving certain entropies and Fenchel–Legendre transforms. Consequences are drawn for asymptotic independence statements and laws of large numbers. In the special case related to quantum physics, our rate function for the occupation measures turns out to be equal to the well-known Donsker–Varadhan rate function for the occupation measures of one motion in the limit of diverging time. This enables us to prove a simple formula for the large-N asymptotic of the symmetrised trace of \({\rm e}^{-\beta {\mathcal{H}}_N}\) , where \({\mathcal{H}}_N\) is an N-particle Hamilton operator in a trap.

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References

  1. Adams S. (2001). Complete equivalence of the Gibbs ensembles for one-dimensional Markov systems. J. Stat. Phys. 105(5/6): 879–908

    Article  MathSciNet  MATH  Google Scholar 

  2. Adams, S.: Large deviations for empirical path measures in cycles of integer partitions, preprint arXiV: math.PR /0702053 (2007)

  3. Adams, S.: Interacting Brownian bridges and probabilistic interpretation of Bose–Einstein condensation, Habilitation thesis, University of Leipzig (in preparation, 2007)

  4. Adams S., Bru J.-B. and König W. (2006). Large deviations for trapped interacting Brownian particles and paths, preprint (2004). Ann. Probab. 34(4): 1340–1422

    Article  MATH  Google Scholar 

  5. Adams S., Bru J.-B. and König W. (2006). Large systems of path-repellent Brownian motions in a trap at positive temperature. Electron. J. Probab. 11: 460–485

    Article  MathSciNet  MATH  Google Scholar 

  6. Adams, S., Dorlas, T.: Asymptotic Feynman–Kac formulae for large symmetrised systems of random walks, preprint arXiV:math-ph/0610026, to appear in Annales de l’institut Henri Poincaré (B) Probabilités et Statistiques (2007)

  7. Ceperley D.M. (1995). Path integrals in the theory of condensed Helium. Rev. Mod. Phys. 67: 279–355

    Article  Google Scholar 

  8. Cornu F. (1996). Correlations in quantum plasmas. Phys. Rev. E 53: 4562–4594

    Article  MathSciNet  Google Scholar 

  9. Dawson D.A. and Gärtner J. (1994). Multilevel large deviations and interacting diffusions. Probab. Theory Relat. Fields 98: 423–487

    Article  MathSciNet  MATH  Google Scholar 

  10. Dembo A. and Zeitouni O. (1998). Large Deviations Techniques and Applications, 2nd edn. Springer, New York

    Book  MATH  Google Scholar 

  11. Deuschel, J.-D., Stroock, D.W.: Large Deviations, AMS Chelsea Publishing, American Mathematical Society (2001)

  12. Dinwoodie I.H. and Zabell S.L. (1992). Large deviations for exchangeable random vectors. Ann. Probab. 20: 1147–1166

    Article  MathSciNet  MATH  Google Scholar 

  13. Donsker M.D. and Varadhan S.R.S. (1975). Asymptotic evaluation of certain Markov process expectations for large time, I. Commun. Pure Appl. Math. 28: 1–47

    Article  MathSciNet  MATH  Google Scholar 

  14. Donsker M.D. and Varadhan S.R.S. (1975). Asymptotic evaluation of certain Markov process expectations for large time, II. Commun. Pure Appl. Math. 28: 279–301

    Article  MathSciNet  MATH  Google Scholar 

  15. Donsker M.D. and Varadhan S.R.S. (1976). Asymptotic evaluation of certain Markov process expectations for large time, III. Commun. Pure Appl. Math. 29: 389–461

    Article  MathSciNet  MATH  Google Scholar 

  16. Donsker M.D. and Varadhan S.R.S. (1983). Asymptotic evaluation of certain Markov process expectations for large time, IV. Commun. Pure Appl. Math. 36: 183–212

    Article  MathSciNet  MATH  Google Scholar 

  17. Feynman R.P. (1953). Atomic theory of the λ transition in Helium. Phys. Rev. 91: 1291–1301

    Article  MATH  Google Scholar 

  18. Föllmer, H.: Random fields and diffusion processes. Ecole d’Eté de Saint Flour XV-XVII. Lecture Notes in Mathematics vol. 1362, pp. 101–203, Springer, Heidelberg (1988)

  19. Föllmer H. and Gantert N. (1997). Entropy minimization and Schrödinger processes in infinite dimensions. Ann. Probab. 25(2): 901–926

    Article  MathSciNet  MATH  Google Scholar 

  20. Gärtner J. (1977). On large deviations from the invariant measure. Theory Probab. Appl. 22(1): 24–39

    Article  MathSciNet  MATH  Google Scholar 

  21. Georgii H.-O. (1988). Gibbs Measures and Phase Transitions. Walter de Gruyter, Berlin

    Book  MATH  Google Scholar 

  22. Ginibre J. (1971). Some applications of functional integration in statistical mechanics and field theory. In: C. de Witt, R. Storaeds (eds), pp. 327–427, Gordon and Breach, New York

  23. den Hollander, F.: Large Deviations. Fields Institute Monographs, AMS (2000)

  24. König W. and Mörters P. (2002). Brownian intersection local times: upper tail asymptotics and thick points. Ann. Probab. 30(4): 1605–1656

    Article  MathSciNet  MATH  Google Scholar 

  25. Nagasawa, M.: Schrödinger Equations and Diffusion Theory. Birkhäuser Basel (1993)

  26. Penrose O. and Onsager L. (1956). Bose-Einstein condensation and liquid Helium. Phys. Rev. 104: 576–584

    Article  MATH  Google Scholar 

  27. Revuz D. and Yor M. (1999). Continuous Martingales and Brownian Motion. Springer, Berlin

    Book  MATH  Google Scholar 

  28. Schrödinger E. (1931). Über die Umkehrung der Naturgesetze. Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. 1931(8/9): 144–153

    MATH  Google Scholar 

  29. Seiringer R. (2003). Ground state asymptotics of a dilute, rotating gas. J. Phys. A Math. Gen. 36: 9755–9778

    Article  MathSciNet  MATH  Google Scholar 

  30. Sütõ A. (1993). Percolation transition in the Bose gas. J. Phys. A Math. Gen. 26: 4689–4710

    Article  MathSciNet  Google Scholar 

  31. Sütõ A. (2002). Percolation transition in the Bose gas: II. J. Phys. A Math. Gen. 35: 6995–7002

    Article  MathSciNet  MATH  Google Scholar 

  32. Sznitman A.S. (1998). Brownian Motion, Obstacles and Random Media. Springer, Berlin

    Book  MATH  Google Scholar 

  33. Tóth B. (1990). Phase Transition in an Interacting Bose System. An Application of the Theory of Ventsel’ and Freidlin. J. Stat. Phys. 61(3/4): 749–764

    Article  MathSciNet  Google Scholar 

  34. Trashorras J. (2002). Large deviations for a triangular array of exchangeable random variables. Ann. Inst. H. Poincaré Probab. Stat. 38(5): 649–680

    Article  MathSciNet  MATH  Google Scholar 

  35. Trashorras, J.: Large deviations for symmetrised empirical measures, preprint available at http://www.ceremade.dauphine.fr/preprints/CMD/2006-55.dvi (2007)

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Authors and Affiliations

  1. Max-Planck Institute for Mathematics in the Sciences, Inselstraße 22-26, 04103, Leipzig, Germany

    Stefan Adams

  2. Mathematisches Institut, Universität Leipzig, Augustusplatz 10/11, 04109, Leipzig, Germany

    Wolfgang König

Authors
  1. Stefan Adams
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  2. Wolfgang König
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Corresponding author

Correspondence to Stefan Adams.

Additional information

Partially supported by DFG grant AD 194/1-1 and by the DFG-Forschergruppe 718 “Analysis and stochastics in complex physical systems”.

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Adams, S., König, W. Large deviations for many Brownian bridges with symmetrised initial-terminal condition. Probab. Theory Relat. Fields 142, 79–124 (2008). https://doi.org/10.1007/s00440-007-0099-5

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  • Received: 30 March 2006

  • Revised: 03 July 2007

  • Published: 06 September 2007

  • Issue Date: September 2008

  • DOI: https://doi.org/10.1007/s00440-007-0099-5

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Keywords

  • Brownian motions
  • Symmetrised distribution
  • Large deviations
  • Occupation measure

Mathematics Subject Classification (2000)

  • 60F10
  • 60J65
  • 82B10
  • 81S40
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