Предельная теорема для решений дифференциальных уравнений со случайной правой частью

R. Z. Khas'minskii

Teoriâ veroâtnostej i ee primeneniâ, Tome 11 (1966) no. 3, pp. 444-462

Résumé

The asymptotic behaviour of the solution $X_\varepsilon(t,\omega)$ of equation (0.1) as $\varepsilon\to0$ is considered. The main assumptions are the following ones: 1) condition (1.1) is fulfilled and the processes $F^{(i)}(x,t,\omega)$ satisfy Ibragirnov's mixing condition (1.5) with $T^6\beta(T)\downarrow0$ as $T\to\infty$, 2) limits (1.4) exist and $\overline\Phi^0(x)\equiv0$. The weak convergence of the process $X_\varepsilon(\tau,\omega)$ $(\tau=\varepsilon^2t)$ to a Markov process $X_0(\tau,\omega)$ is proved. Moreover the local characteristics of the process $X_0(\tau,\omega)$ are calculated. An application of this theorem to the problem of parametric excitation of linear systems by random forces is considered