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

In this article, a notion of bi-spatial continuous random dynamical system 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 $ \mathbb{R}^N $ with additive white noise and a polynomial-like growth nonlinearity of order $ p, p\geq2 $. We prove that this equation generates a bi-spatial $ (L^2(\mathbb{R}^N), H^s(\mathbb{R}^N)\cap L^p(\mathbb{R}^N)) $-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 $ H^s(\mathbb{R}^N)\cap L^p(\mathbb{R}^N) $, where $ H^s(\mathbb{R}^N) $ is a fractional Sobolev space with $ s\in(0,1) $. Especially, the measurability of pullback attractor is individually derived by proving the the continuity of solutions in $ H^s(\mathbb{R}^N) $ and $ L^p(\mathbb{R}^N) $ 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 $ H^s(\mathbb{R}^N)\cap L^p(\mathbb{R}^N),s\in(0,1),N\geq1 $.