Scientific article
Open access

Long time behavior of the solutions to non-linear Kraichnan equations

Published inProbability theory and related fields, vol. 131, no. 4, p. 493-518
Publication date2005

We consider the solution of a nonlinear Kraichnan equation $$partial_s H(s,t)=int_t^s H(s,u)H(u,t) k(s,u) du,quad sge t$$ 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.

  • arxiv : math.PR
Citation (ISO format)
GUIONNET, Alice, MAZZA, Christian. Long time behavior of the solutions to non-linear Kraichnan equations. In: Probability theory and related fields, 2005, vol. 131, n° 4, p. 493–518. doi: 10.1007/s00440-004-0382-7
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Article (Accepted version)
ISSN of the journal0178-8051

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