Caputo mean-square attractors for non-autonomous stochastic differential equations

Abstract

This paper investigates Caputo mean-square attractors for non-autonomous stochastic evolution systems. We first introduce the concept of Caputo mean-square attractors and then establish a sufficient criterion for existence of such attractors.As an application, we consider a non-autonomous Caputo fractional stochastic differential equation of order $\alpha\in (\frac{1}{2},1)$ in $L^2(\Omega; \mathbb{R}^d)$ with a driving system on a compact base space $P$ and tempered fractional noise.It is shown that this equation generates a Caputo mean-square random semi-dynamical system on $\mathfrak{C} \times P$ with a skew-product semi-flow structure,where $\mathfrak{C}$ denotes the space of continuous functions $f\in \mathbb{R}^{+}\rightarrow L^2(\Omega; \mathbb{R}^d)$. Under suitable conditions, we prove that this semi-dynamical system admits a Caputo mean-square attractor.