Asymptotic $ H^2$ regularity of a stochastic reaction-diffusion equation

In this paper we study the asymptotic dynamics for the weak solutions of the following stochastic reaction-diffusion equation defined on a bounded smooth domain \begin{document}$ {\mathcal{O}} \subset {\mathbb{R}}^N $\end{document} , \begin{document}$ N \leqslant 3 $\end{document} , with Dirichlet boundary condition:\begin{document}$ \begin{equation} \nonumber\begin{aligned} { {{\rm{d}}} u } +(-\Delta u + u ^3- \beta u ) {{\rm{d}}} t = g(x) {{\rm{d}}} t+h(x) {{\rm{d}}} W , \quad u|_{t = 0} = u_0\in H: = L^2( {\mathcal{O}}), \end{aligned} \end{equation} $\end{document}where \begin{document}$ \beta>0 $\end{document} , \begin{document}$ g\in H $\end{document} , and \begin{document}$ W $\end{document} a scalar and two-sided Wiener process with a regular perturbation intensity \begin{document}$ h $\end{document} . We first construct an \begin{document}$ H^2 $\end{document} tempered random absorbing set of the system, and then prove an \begin{document}$ (H,H^2) $\end{document} -smoothing property and conclude that the random attractor of the system is in fact a finite-dimensional tempered random set in \begin{document}$ H^2 $\end{document} and pullback attracts tempered random sets in \begin{document}$ H $\end{document} under the topology of \begin{document}$ H^2 $\end{document} . The main technique we shall employ is comparing the regularity of the stochastic equation to that of the corresponding deterministic equation for which the asymptotic \begin{document}$ H^2 $\end{document} regularity is already known.