Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential Equations

Abstract

This work is concerned with the quadratic-mean asymptotically almost periodic mild solutions for a class of stochastic functional differential equations $\text{d}x\left(t\right)=\left[A\left(t\right)x\left(t\right)+F\left(t,x\left(t\right),x_t\right)\right]\text{d}t+H(t,x\left(t\right),x_t)\circ \text{d}W(t)$. A new criterion ensuring the existence and uniqueness of the quadratic-mean asymptotically almost periodic mild solutions for the system is presented. The condition of being uniformly exponentially stable of the strongly continuous semigroup ${\{T\left(t\right)\}}_{t\ge 0}$ is essentially removed, which is generated by the linear densely defined operator $A:D(A)\subset L^2(\mathbb{P},ℍ)\to L^2(\mathbb{P},ℍ)$, only using the exponential trichotomy of the system, which reflects a deeper analysis of the behavior of solutions of the system. In this case the asymptotic behavior is described through the splitting of the main space into stable, unstable, and central subspaces at each point from the flow’s domain. An example is also given to illustrate our results.