Probability Theory and Related Fields

, Volume 131, Issue 4, pp 493–518

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

  • Alice Guionnet
  • Christian Mazza

DOI: 10.1007/s00440-004-0382-7

Cite this article as:
Guionnet, A. & Mazza, C. Probab. Theory Relat. Fields (2005) 131: 493. doi:10.1007/s00440-004-0382-7


We consider the solution of a nonlinear Kraichnan equation Open image in new window 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Alice Guionnet
    • 1
  • Christian Mazza
    • 2
  1. 1.Ecole Normale Supérieure de LyonUnité de Mathématiques pures et appliqueéesLyon Cedex 07France
  2. 2.Section de MathématiquesGenève 24Suisse