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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We are very grateful to P. Gerard and R. Speicher for cheerful and motivating discussions.
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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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DOI: https://doi.org/10.1007/s00440-004-0382-7