Starting from the cumulant semigroup of a measure-valued branching process, we construct the transition probabilities of some Markov process Y (β) = (Yt (β), t ∈ R), which we call a measure-valued branching process with discrete immigration of unitβ. The immigration of Y (β) is governed by a Poisson random measure р on the time-distribution space and a probability generating function h; both depending on β. It is shown that, under suitable hypotheses, Y (β) approximates to a Markov process Y = (Yt, t ∈ R) as β→ 0+. The latter is the one we call a measure-valued branching process with immigration. The convergence of branching particle systems with immigration is also studied.

The immigration processes associated with a given branching particle system are formulated by skew convolution semigroups. It is shown that every skew convolution semigroup corresponds uniquely to a locally integrable entrance law for the branching particle system. The immigration particle system may be constructed using a Poisson random measure based on a Markovian measure determined by the entrance law. In the special case where the underlying process is a minimal Brownian motion in a bounded domain, a general representation is given for locally integrable entrance laws for the branching particle system. The convergence of immigration particle systems to measure-valued immigration processes is also studied.

The immigration structure associated with a measure-valued branching process may be described by a skew convolution semigroup. For a special type of measure-valued branching process, the Dawson-Watanabe superprocess, we show that a skew convolution semigroup corresponds uniquely to an infinitely divisible probability measure on the space of entrance laws for the underlying process. An immigration process associated with a Borel right superprocess does not always have a right continuous realization, but it can always be obtained by transformation from a Borel right one in an enlarged state space.

We prove a central limit theorem for the super-Brownian motion with immigration governed by another super-Brownian. The limit theorem leads to Gaussian random fields in dimensions d≥3. For d = 3 the field is spatially uniform; for d≥5 its covariance is given by the potential operator of the underlying Brownian motion; and for d = 4 it involves a mixture of the two kinds of fluctuations.