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

Abstract

We consider the solution of a nonlinear Kraichnan equation $${\partial }_{s}H(s,t)={\int }_t^{s}H(s,u)H(u,t)k(s,u)du,s\ge t$$ 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.

32 pages

References in corpus (1)

• Dynamics of glassy systems