A General Family of Branching Markov Processes

Abstract

Recall that, in Chap. 2, we introduced the notion of a general Markov process on E which is taken to be a locally compact Hausdorff space, to which we can append a cemetery state, $\dagger$. We used the notation $\mathtt P=(\mathtt P_t, t\geq 0)$ to denote its associated semigroup and, accordingly, we later referred to it as a $\mathtt P$-Markov process. As a generalisation of the neutron branching processes discussed in Chap. 3, we are interested in spatial branching processes that are defined in terms of a $\mathtt P$-Markov process and a branching operator. In this chapter and subsequent chapters, we introduce such processes and develop a number of generic results for them. Some of the notations used for neutron branching processes (NBPs) will also be used in this general setting. This is deliberate to give the reader a chance to see how the former is an interesting core example of the latter.