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Long time behaviour of the solution to non-linear Kraichnan equations
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  • Published: 11 November 2004

Long time behaviour of the solution to non-linear Kraichnan equations

  • Alice Guionnet1 &
  • Christian Mazza2 

Probability Theory and Related Fields volume 131, pages 493–518 (2005)Cite this article

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  • 5 Citations

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Abstract.

We consider the solution of a nonlinear Kraichnan equation with a covariance kernel k and boundary condition H(t, t)=1. We study the long time behaviour of H as the time parameters t, s go to infinity, according to the asymptotic behaviour of k. This question appears in various subjects since it is related with the analysis of the asymptotic behaviour of the trace of non-commutative processes satisfying a linear differential equation, but also naturally shows up in the study of the so-called response function and aging properties of the dynamics of some disordered spin systems.

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References

  1. Abramowitz, Stegun,: Handbook of mathematical functions with formulas. graphs and mathematical tables Dover, 1992

  2. Ben Arous, G., Dembo, A., Guionnet, A.: Aging of spherical spin glasses. Prob. Theory. Relat. Fields. 120, 1–67 (2001)

    Google Scholar 

  3. Ben Arous, G., Dembo, A., Guionnet, A.: Limiting dynamics for the Langevin dynamics of p-spins models. Preprint 2004

  4. Brak, R.T., Prellberg,: Critical exponents from non-linear functional equations for partially directed cluster models. http://www.ms.unimelb.edu.au/ brak/publications.html

  5. Cartan, H.: Théorie élémentaire des fonctions analytiques d’une ou plusieurs variables complexes. Hermann, 1978

  6. Chandra, P., Feigleman, M.V., Ioffe, L., Kagan, D.: History-Dependence and Ageing in a Periodic Long-Range Josephson Array. Phys. Rev. B 56, 11553 (1997)

    Article  Google Scholar 

  7. Coulomb, J.: Sur les zéros des fonctions de Bessel considérées comme fonction de l’ordre. Bull. Sci. Math. 60, 297–302 (1936)

    MATH  Google Scholar 

  8. Cugliandolo, L., Kurchan, J.: Analytical Solution of the Off-Equilibrium Dynamics of a Long Range Spin-Glass Model. Phys. Rev. Lett. 71, 173 (1993)

    Article  Google Scholar 

  9. Cugliandolo, L.: Dynamics of glassy systems. Les Houches, 2002

  10. Erdelyi,: Higher transcendental functions. Bateman manuscript project. California Institute of Technology, vol. 2, 1953, section 7.9, pp. 61

  11. Franz, S., Hertz, J.: Glassy transition and aging in a model without disorder. Phys. Rev. Lett. 74, 2114 (1995)

    Article  Google Scholar 

  12. Frisch, U., Bourret, R.: Parastochastics J. Math. Phys. 11, 364 (1970)

    Article  MATH  Google Scholar 

  13. Kraichnan, R.: Dynamics of Nonlinear Stochastic systems. Jour. Math. Phys. 2, 124 (1961)

    MATH  Google Scholar 

  14. Neu, P., Speicher, R.: A self-consistent master equation and a new kind of cumulants. Zeitschrift fur Physik B 92, 399 (1993)

    Google Scholar 

  15. Voiculescu, D.: Lectures on Free Probability. In: Lectures Notes in Mathematics 1738, Springer, 2000

  16. Widder, D.: The Laplace Transform. Princeton University Press, 1946

  17. Watson, G.: A treatrise on the Theory of Bessel functions. Cambridge University Press, 1966

  18. Watson, G. A course in Modern Analysis. Fourth Edition. Cambridge University Press, 1927

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Acknowledgments.

We are very grateful to P. Gerard and R. Speicher for cheerful and motivating discussions.

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Authors and Affiliations

  1. Ecole Normale Supérieure de Lyon, Unité de Mathématiques pures et appliqueées, UMR 5669, 46 Allée d’Italie, 69364, Lyon Cedex 07, France

    Alice Guionnet

  2. Section de Mathématiques, 2-4 Rue du Lièvre, CP 240, CH-1211, Genève 24, Suisse

    Christian Mazza

Authors
  1. Alice Guionnet
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  2. Christian Mazza
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Guionnet, A., Mazza, C. Long time behaviour of the solution to non-linear Kraichnan equations. Probab. Theory Relat. Fields 131, 493–518 (2005). https://doi.org/10.1007/s00440-004-0382-7

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  • Received: 05 March 2003

  • Revised: 17 June 2004

  • Published: 11 November 2004

  • Issue Date: April 2005

  • DOI: https://doi.org/10.1007/s00440-004-0382-7

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Keywords

  • Boundary Condition
  • Differential Equation
  • Covariance
  • Stochastic Process
  • Asymptotic Behaviour
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