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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 $\mathfrak{C}_{w}=C(\mathbb{R}^{+}, L^{2}_w(\Omega, \mathcal{F}; \mathbb{H}))$ 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.

Submission history

From: Lijuan Zhang [view email]
[v1] Sat, 14 Feb 2026 02:37:11 UTC (17 KB)
[v2] Fri, 10 Jul 2026 05:36:50 UTC (318 KB)