A limit theorem for solutions of differential equations with random right hand side

A. N. Borodin

Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 3, pp. 498-512

Résumé

The main purpose of this paper is to weak requirements in a theorem of Has'minski\u i [2]. The asymptotic behaviour of the solution $X_{\varepsilon}(t,\omega)$ of equation (0.1) as $\varepsilon\to 0$ is studied. The main assumptions are the following: conditions (1.1) and (1.2) are fulfilled, the processes $F^{(i)}(x,t,\omega)$ satisfy Kolmogorov's mixing condition (0.4) (for a special type of processes $F^{(i)}$, see condition (4'), Rosenblatt's mixing condition (0.3) is sufficient), limits (1.4) and (1.5) exist. Under these assumptions and some additional ones the process $X_{\varepsilon}(\tau/\varepsilon^2,\omega)$ is proved to converge weakly to a Markov process $X_0(\tau,\omega)$. The local characteristics of $X_0(\tau,\omega)$ are calculated from condition (1.5).