LAUSR

Abstract We investigate the regularity of random attractors for the non-autonomous non-local fractional stochastic reaction–diffusion equations in H s ( R n ) with s ∈ ( 0 , 1 ) . We prove the existence and uniqueness of the tempered random attractor that is compact in H s ( R n ) and attracts all tempered random subsets of L 2 ( R n ) with respect to the norm of H s ( R n ) . The main difficulty is to show the pullback asymptotic compactness of solutions in H s ( R n ) 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.