Stationary Distributions for Stochastic Evolution Equations with Non-dense Domain

In this paper, we consider the semilinear stochastic evolution equation with a non-dense domain. Using the theory of integrated semigroups and the resolvent operator, we study the existence and uniqueness of the mild solution by a modified variation of constants formula. The relation between the mild solution and the strong solution is established. We show that the mild solution to the non-densely defined stochastic evolution equation is a Markov process and a Feller process. Under some conditions, we investigate the ultimate boundedness in mean square of the mild solution and the asymptotic closeness of mild solutions starting at two different initial values. By virtue of the two properties, the sufficient conditions for the existence and uniqueness of the stationary distribution are derived. Finally, two examples of a stochastic age-structured model and a stochastic parabolic equation are presented to demonstrate the application of our main results.