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Stability of a random diffusion with linear drift. (English) Zbl 0856.93102
The authors investigate the long term behavior of the solutions of an SDE with random drift and diffusion coefficients (depending on a finite state Markov process as background noise which is independent of the driving Wiener process). Under a condition which involves the drift and diffusion coefficients as well as the jump rates of the Markov process, with the help of Lyapunov type functions they show the following. The two-point flow generated by the SDE is shown to be stable in the $$p$$th mean; a fact which is shown to be implied, essentially, by the mean square stability of the associated linear system driven by the Markov noise as real noise (with diffusion = zero). The transition probabilities of the pair solution/Markov noise converge weakly to an invariant measure.

##### MSC:
 93E15 Stochastic stability in control theory 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J27 Continuous-time Markov processes on discrete state spaces
##### Keywords:
stability; Lyapunov functions; finite state Markov process
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