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

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

2004-09-16

arxiv.org/abs/math/0409274v1

Mathematics

Probability

32 pages

Scientific paper

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

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