MEASURE-VALUED BRANCHING MARKOV PROCESSES
Zenghu Li
Measure-valued branching processes constitute a class of Markov processes with many beautiful mathematical structures and interesting applications. These processes appeared in Jirina (1958, 1964) and Watanabe (1968) as high density limits of branching particle systems. Their connection with stochastic evolution equations was investigated in Dawson (1975). A special class of measure-valued branching processes, known as Dawson-Watanabe processes, have been studied extensively in the past decades. The development of this subject has been stimulated from different subjects including branching processes, interacting particle systems, stochastic partial differential equations and non-linear partial differential equations. The study of those processes has also led to better understanding of results in those subjects; see Dawson (1992, 1993), Dynkin (1994), Etheridge (2000) and Le Gall (1999).
A branching process describes the evolution of a population evolving randomly in a isolated region. A useful and realistic modification of this scheme is the addition of immigration into the population from an outside source. Clearly, branching models allowing mmigration are of great importance from the point of view of applications; see, e.g., Athreya and Ney (1972, p.10 and p.262). The modification is also familiar from the literature of superprocesses; see, e.g., Dawson (1993), Dawson and Ivanoff (1978) and Dynkin (1991). We here adopt an axiomatic formulation of the immigration processes suggested in Li (1995/6), which gives the general formulation of the structures of immigration independent of the inner population.
Needless to say, most of the theory of Dawson-Watanabe superprocesses carries over to their associated immigration processes and could be developed by techniques very close to those in Dawson (1992, 1993). However, the immigration superprocesses also involve many new structures. We shall see that most of the fundamental problems on immigration superprocesses have succinct solutions in terms of skew convolution semigroups and related concepts, especially those from probabilistic potential theory. The recent development shows that skew convolution semigroups also play some roles in the study of generalized Ornstein-Uhlenbeck processes and affine Markov processes; see, e.g., Bogachev et al (1996), Dawson and Li (2004, 2006) and Dawson et al (2004).
These notes originated from lectures on Dawson-Watanabe superprocesses and immigration superprocesses the author gave at several places. Our purpose is to give a compact and self-contained treatment of the constructions and basic properties of those processes. For the convenience of the reader, in the last chapter we give a summary of some results from the general theory of Markov processes which is needed in the main text. Most of the results can be found in Sharpe (1988). We hope that this treatment may lead the reader to a quick appreciation of the original work.
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