Mean-square attractors for non-autonomous Caputo fractional stochastic differential equations

Abstract

This paper investigates the existence of mean-square attractors for a class of non-autonomous Caputo fractional stochastic differential equations of order $\alpha \in (\frac{1}{2},1)$, with a driving system on a compact base space $P$ and tempered fractional noise. We first construct a mean-square semi-dynamical system on $\mathfrak{C}\times P$ that carries a skew-product semi-flow structure, where $\mathfrak{C}=C({\mathbb{R}}^{+},L^2(\Omega ,\mathcal{F};{\mathbb{R}}^d))$ denotes the space of continuous functions from ${\mathbb{R}}^{+}$ into $L^2(\Omega ,\mathcal{F};{\mathbb{R}}^d)$. A global forward attracting set is then established in the weak mean-square topology. Moreover, by endowing the function space ${\mathfrak{C}}_w=C({\mathbb{R}}^{+},L_w^2(\Omega ,\mathcal{F};\mathbb{H}))$ with an appropriate topology that renders it complete, we show that the skew-product semi-flow possesses a bounded and closed mean-square attractor within ${\mathfrak{C}}_w\times P$. It is worth emphasizing that completeness plays a crucial role here: without this property, the attractor need not exist.