Qualitative behaviour of stochastic delay equations with a bounded memory

Abstract

A dichotomy is proved concerning recurrence properties of the solution of certain stochastic delay equations. If the solution process is recurrent, there exists an invariant measure π on the state space C which is unique (up to a multiplicative constant) and the tail-field is trivial. If π happens to be a probability measure, then for every initial condition, the distribution of the process converges to it as t→∞. We will formulate a sufficient condition for the existence of an invariant probability measure (ipm) in icrnia of Lyapunov junctionals and give two examples, one Heing the stochastic-delay version of the famous logistic equation of population growth. Finally we study approximations of delay equations by Markov chains.