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.


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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