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Qualitative behaviour of stochastic delay equations with a bounded memory. (English) Zbl 0539.60061
A dichotomy is proved concerning recurrence properties of the solution of certain stochastic delay equations. If the solution process is recurrent, there exists a \(\sigma\)-finite invariant measure \(\pi\) on the state space C which is unique (up to a multiplicative constant) and the tail- \(\sigma\)-field is trivial. If \(\pi\) happens to be a probability measure, then for every initial condition, the law of the process converges to it as \(t\to \infty.\)
We formulate a sufficient condition for the existence of an invariant probability measure in terms of Lyapunov functionals and give two examples, one being the stochastic-delay version of the logistic equation of population growth. Finally we study approximations of delay equations by Markov chains.

MSC:
60H25 Random operators and equations (aspects of stochastic analysis)
92D25 Population dynamics (general)
60J25 Continuous-time Markov processes on general state spaces
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
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