ABSTRACT
In this section we further develop the semigroup analysis of the intensity measures of spatial branching processes discussed in Section 6.1 We consider a sequence of nonnegative measures µn and a collection of nonnegative potential functions Gn on some measurable state spaces En. We also let Mn be some Markov transitions from En−1 into En. We associate with these objects the flow of nonnegative measures
γn+1 = γnQn+1 + µn+1 (13.1)
with the initial condition γ0 = µ0, and the integral operator Qn+1 from En into En+1 is defined by
Qn+1(xn, dxn+1) := Gn(xn) Mn+1(xn, dxn+1) (13.2)
These distributions can be interpreted as the intensity measures of a spatial branching process. We refer the reader to Chapter 6 for a detailed discussion on these spatial point processes.
