Stability properties of solutions of linear second order differential equations with random coefficients

The equation is considered where is given and is a random sequence. Sufficient conditions are proved which guarantee either stability or instability for the zero solution. Stability means that all solutions almost surely tend to zero as t→∞. By instability we mean that the sequence of the expected values of the amplitudes of every solution tends to infinity as k→∞. It turns out that ak ∞ (k→∞) implies stability for all absolutely continuous distributions and for the “overwhelming majority” of the singular distributions. The instability theorem is applied to the problem of random swinging, when is periodic with two different terms (Meissnerʹs equation) and are independent identically distributed random variables. The application gives conditions for stochastic parametric resonance.