Stochastic Stability of Some Mechanical Systems

We discuss the behavior, for large values of time, of two linear stochastic mechanical systems. The systems are similar mathematically in that they contain a white noise in their parameters. The initial data may be random as well but are independent of white noise. The expected energy is calculated in both cases. It is well known that for free nonstochastic mechanical systems with viscous damping, the energy approaches zero as time increases. We check that this behavior takes place for the stochastic systems under consideration in the case when the initial data are random but the parameters are not. When the parameters contain a random noise the expected energy may be infinite, approach zero, remain bounded, or increase with no bound. This regime is similar to but more interesting than the known regime for the solutions of differential equations with time dependent periodic coefficients that describes the behavior of a mechanical system with characteristics that are periodic functions of time. We give necessary and sufficient conditions for stability of both systems in terms of the structure of the set of roots of an auxiliary equation.