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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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