Long time behaviour of the solution to non-linear Kraichnan equations
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- Guionnet, A. & Mazza, C. Probab. Theory Relat. Fields (2005) 131: 493. doi:10.1007/s00440-004-0382-7
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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.