We consider a branching particle system where an individual particle gives birth to a random number of offspring at the place where it dies. The probability distribution of the number of offspring is given by pk, k = 2, 3, …. The corresponding branching process is related to the semilinear partial differential equation for x ∈ ℝd, where A is the infinitesimal generator of a multiplicative semigroup and the pks, k = 2, 3, …, are nonnegative functions such that We obtain sufficient conditions for the existence of global positive solutions to semilinear equations of this form. Our results extend previous work by Nagasawa and Sirao (1969) and others.

Using a representation in terms of a two-type branching particle system, we prove that positive solutions of the system remain bounded for suitable bounded initial conditions, provided A and B generate processes with independent increments and one of the processes is transient with a uniform power decay of its semigroup. For the case of symmetric stable processes on R1,this answers a question raised in [4].

In 1991 Perkins [7] showed that the normalized critical binary branching process is a time inhomogeneous Fleming-Viot process. In the present paper we extend this result to jump-type branching processes and we show that the normalized jump-type branching processes are in a new class of probability measure-valued processes which will be called ‘jump-type Fleming-Viot processes’. Furthermore we also show that by using these processes it is possible to introduce another new class of measure-valued processes which are obtained by the combination of jump-type branching processes and Fleming-Viot processes.

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.

We obtain a representation for the supercritical Dawson-Watanabe superprocessin terms of a subcritical superprocess with immigration, where the immigration at a given time is governed by the state of an underlying branching particle system. The proof requires a new result on the laws of weighted occupation times for branching particle systems.