A typical class of generalized OU-processes arise as small-branching fluctuation limits of subcritical immigration superprocesses around their equilibrium means. In this chapter, we first establish such a fluctuation limit theorem in the space of Schwartz distributions. A stronger result is then proved which shows that the convergence actually holds in a suitable weighted Sobolev space. To avoid complicated regularity assumptions, we only consider the case where the spatial motion is a Brownian motion with killing.