Probability Theory and Related Fields

, Volume 131, Issue 4, pp 493-518

First online:

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

  • Alice GuionnetAffiliated withEcole Normale Supérieure de Lyon, Unité de Mathématiques pures et appliqueées
  • , Christian MazzaAffiliated withSection de Mathématiques

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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.