Abstract
A measure-valued process describes the evolution of a population that evolves according to the law of chance. In this chapter we provide some basic characterizations and constructions for measure-valued branching processes. In particular, we establish a one-to-one correspondence between those processes and cumulant semigroups. Some results for nonlinear integral evolution equations are proved, which lead to an analytic construction of a class of measure-valued branching processes, the socalled Dawson–Watanabe superprocesses.We shall construct the superprocesses for admissible killing densities and general branching mechanisms that are not necessarily decomposable into local and non-local parts. A number of moment formulas are proved.We also give some estimates for the variations of the transition probabilities with different initial states in Wasserstein and total variation distances.
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