Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain

Abstract

<p style='text-indent:20px;'>In this article, a notion of <i>bi-spatial continuous random dynamical system</i> is introduced between two completely separable metric spaces. It is show that roughly speaking, if such a random dynamical system is asymptotically compact and random absorbing in the initial space, then it admits a bi-spatial pullback attractor which is measurable in two spaces. The measurability of pullback attractor in the regular spaces is completely solved theoretically. As applications, we study the dynamical behaviour of solutions of the non-autonomous stochastic fractional power dissipative equation on <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> with additive white noise and a polynomial-like growth nonlinearity of order <inline-formula><tex-math id="M2">\begin{document}$ p, p\geq2 $\end{document}</tex-math></inline-formula>. We prove that this equation generates a bi-spatial <inline-formula><tex-math id="M3">\begin{document}$ (L^2(\mathbb{R}^N), H^s(\mathbb{R}^N)\cap L^p(\mathbb{R}^N)) $\end{document}</tex-math></inline-formula>-continuous random dynamical system, and the random dynamics for this system is captured by a bi-spatial pullback attractor which is compact and attracting in <inline-formula><tex-math id="M4">\begin{document}$ H^s(\mathbb{R}^N)\cap L^p(\mathbb{R}^N) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ H^s(\mathbb{R}^N) $\end{document}</tex-math></inline-formula> is a fractional Sobolev space with <inline-formula><tex-math id="M6">\begin{document}$ s\in(0,1) $\end{document}</tex-math></inline-formula>. Especially, the measurability of pullback attractor is individually derived by proving the the continuity of solutions in <inline-formula><tex-math id="M7">\begin{document}$ H^s(\mathbb{R}^N) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ L^p(\mathbb{R}^N) $\end{document}</tex-math></inline-formula> with respect to the sample. A difference estimates approach, rather than the usual truncation estimate and spectral decomposition technique, is employed to overcome the loss of Sobolev compact embedding in <inline-formula><tex-math id="M9">\begin{document}$ H^s(\mathbb{R}^N)\cap L^p(\mathbb{R}^N),s\in(0,1),N\geq1 $\end{document}</tex-math></inline-formula>.