On the Theory of Differential Equations with Random Coenfficients

G. Ya. Lyubarskii   ; Yu. L. Rabotnikov

Teoriâ veroâtnostej i ee primeneniâ, Tome 8 (1963) no. 3, pp. 309-318

Résumé

The equation $\ddot u(t)+a_1(t)\dot u(t)+[\alpha(t)-\alpha(t)]u(t)=0$ is considered where the coefficient $a_1(t)$ and $a_0 (t)$ are real, piecewise continuous and periodic functions with the same period $T$ and $\alpha (t)$ is a real random function. The restrictions on the $\alpha (t)$ are essentially the following. The correlation length $\alpha $ is much shorter than the period $T$, the random function $\alpha(t)$, $\infty<t<\infty$, does not exceed the value ${\gamma/{\sqrt a(\gamma={\text{const}}<1)}}$. The necessary and sufficient conditions are found for the boundedness of mean values $Mu^2 (t),M[u(t)\dot u(t)]$ and $M\dot u^2 (t)$.