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Stochastic stability of some mechanical systems with a multiplicative white noise. (English) Zbl 1056.60056

Summary: We discuss the behavior, for large values of time, of a class of linear mechanical systems with a multiplicative white noise in its parameters. The initial conditions may be random as well but are independent of white noise. It is well known that a deterministic linear mechanical system with viscous damping is stable, i.e., its energy approaches zero as time increases. We calculate the expected energy and check that this behavior takes place in the case when the initial conditions 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. We give necessary and sufficient conditions for stability of the systems considered in terms of the roots of an auxiliary equation.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
35B35 Stability in context of PDEs
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