Asymptotic analysis of a certain random differential equation (Q760966)

One considers an initial value problem \(dy_{\lambda}/dt=\lambda \cdot V(t)y_{\lambda}\), \(y_{\lambda}(0)=\phi\), in a Hilbert space \((H,<\cdot >)\) with V(t), \(t\geq 0\), being an operator valued process, stationary in time, satisfying some measurability and mixing conditions. [See for comparison: \textit{G. C. Papanicolaou} and \textit{S. R. S. Varadhan}, Commun. Pure Appl. Math. 26, 497-524 (1973; Zbl 0253.60065)]. Here it is shown that in the limit \(\lambda\) \(\to 0\) (''weak coupling limit'') one gets \[ {\mathcal E}<y_{\lambda}(t\cdot \lambda^{-2}),\psi >=<S(t)\phi,\psi > \] for all \(\psi\in H\) and \(t>0\), where S(t), \(t\geq 0\), is a contraction semigroup, whose generator \(\bar V\) is explicitly given by \[ \bar V\psi =\lim_{T\to \infty}T^{-1}\cdot \iint_{0\leq r,s\leq T}dsdr{\mathcal E}V(s)V(r)\psi. \]