Regularity of random attractors for fractional stochastic reaction-diffusion equations on Rn

We investigate the regularity of random attractors for the non-autonomous non-local fractional stochastic reaction-diffusion equations in Hs(Rn) with s∈(0,1). We prove the existence and uniqueness of the tempered random attractor that is compact in Hs(Rn) and attracts all tempered random subsets of L2(Rn) with respect to the norm of Hs(Rn). The main difficulty is to show the pullback asymptotic compactness of solutions in Hs(Rn) due to the noncompactness of Sobolev embeddings on unbounded domains and the almost sure nondifferentiability of the sample paths of the Wiener process. We establish such compactness by the ideas of uniform tail-estimates and the spectral decomposition of solutions in bounded domains.