Fluctuation limit theorems of immigration superprocesses with small branching

We establish fluctuation limit theorems of immigration superprocesses with small branching rates. The weak convergence of the processes on a Sobolev space is established, which improves the result of Gorostiza and Li (High density fluctuations of immigration branching particle systems. In: Gorostiza, L.G., Ivanoff, B.G. (Eds.), Stochastic Models, (Ottawa, Ontario, 1998), CMS Conference Proceedings 2000, Series vol. 26, American Mathematical Society, Providence, RI, pp. 159–171). The limiting processes are infinite-dimensional Ornstein–Uhlenbeck type processes.