Poisson stable solutions and solution maps for stochastic functional differential equations

Abstract

We observe Poisson stable solutions for nonlinear stochastic functional differential equations (SFDEs) with finite delay. Firstly, we prove the existence and uniqueness of bounded (in square-mean sense) solutions and solution maps for SFDEs with finite delay by remoting start (or dissipative method) and classical pull-back attraction method. Then, based on the relationship between the solution and coefficients, we obtain such Poisson stable solutions by using Shcherbakov’s comparability method in character of recurrence. Because the solutions of the delay equations are not-Markov, we employ the solution maps in the appropriate phase space as a viable alternative for studying further asymptotic properties, and we also discuss Poisson stable solution maps and their asymptotic stability. For illustration of our results, we give the application arising from stochastic Lotka–Volterra cooperative systems with distributed delay.