Branching measure-valued processes
The studied object is a general subcritical branching measure-valued process \(X_ t\), that is, a time-inhomogeneous Markov process whose state space consists of finite measures on a Luzin space which satisfies the branching property \[ \log P_{r, \mu} e^{-\langle f, \mu \rangle} = \int \mu (dx) \log P_{r, \delta_ x} e^{-\langle f, \mu \rangle} \] and in addition satisfies certain standard regularity conditions including right continuity and quasi-left continuity, and bounded first- and second-moment conditions. The main result of the paper is that if \[ P_{r, \mu} \langle f, X_ t \rangle = \Pi_{r, \mu} f(\xi_ f), \] where \(\Pi_{r, \mu}\) is the law of a Markov process (with possible finite death time) in \(E\), then such a process can be represented as superprocess with branching characteristics \((Q, l)\) where the continuous branching characteristic \(Q\) is a continuous natural additive functional and the discontinuous branching characteristic \(l (ds, d \nu)\) is such that \(\int l (A,d \nu) \nu (B)\) is dominated by the Lévy measure \(N_ \xi (ds, dy)\) of the process \(\xi\). \(Q\) and \(l\) are obtained in terms of the continuous martingale component of the total mass process \(X_ t (E)\) and the Lévy measure (jumps compensator) of \(X\), respectively. The characteristics \((Q,l)\) are related to the additive functional \(K\) and one parameter family of operators, \(\psi^ s\) in \(p {\mathcal B}\), which were formulated in previous papers of Dynkin. Existence of superprocesses under conditions on \(K\) and \(\psi^ s\) was established by Dynkin.
scientific article; zbMATH DE number 627487