Stability in distribution for stochastic differential equations with memory driven by positive semigroups and Levy processes

In this paper, we consider stationarity of a class of stochastic differential equations with memory driven by Levy processes in Banach spaces. The stochastic systems under investigation have linear operators acting on point or distributed delayed terms and the operators acting on the instantaneous term generate positive strongly continuous semigroups. The asymptotic behavior of the associated deterministic systems is considered through the useful Weis Theorem on a Banach lattice, and the stationarity of the generalized nonlinear stochastic systems is established through a weak convergence programme. Last, our theory is illustrated by its application to a stochastic age-structured population model with memory on which the usual spectral-determined growth condition type of scheme seems to be impossibly carried out. (C) 2019 Elsevier Inc. All rights reserved.