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In this paper, a standard about the existence and upper semi-continuity ofpullback attractors in the non-initial space is established for some classes ofnon-autonomous SPDE. This pullback attractor, which is the omega-limit set ofthe absorbing set constructed in the initial space, is completely determined bythe asymptotic compactness of solutions in both the initial and non-initialspaces. As applications, the existences and upper semi-continuity of pullbackattractors in $H^1(\mathbb{R}^N)$ are proved for stochastic non-autonomousreaction-diffusion equation driven by a multiplicative noise. Finally we showthat under some additional conditions the cocycle admits a unique equilibrium.

Regularity of pullback attractors and equilibria for a stochastic non-autonomous reaction-diffusion equations perturbed by a multiplicative noise