Measure-valued immigration diffusions and generalized Ornstein-Uhlenbeck diffusions

Three different kinds of fluctuation limits (high density fluctuation, small branching fluctuation and large scale fluctuation) of the measure-valued immigration diffusion process are studied, which lead to the generalized Ornstein-Uhlenbeck diffusion defined by a Langevin equation of the type of Holley and Stroock (1978). The fluctuation limit theorems cover all dimen-sion numbers and give physical interpretations to the parameters appearing in the equation