Grey-type conservativity criterion for multitype continuous-state branching processes

Characterize the branching mechanisms of multitype continuous-state branching processes for which the cemetery state is reached in finite time with positive probability, thereby obtaining necessary and sufficient conservativity conditions analogous to Grey’s criterion for one-type continuous-state branching processes.

Background

For a multitype continuous-state branching process, conservativity means that the process does not reach the cemetery state in finite time. The paper explains that, unlike extinction at the absorbing state 0, cemetery-state absorption depends on all components of the multitype branching mechanism and is therefore substantially more difficult to characterize.

The authors establish several necessary and sufficient conditions, including a complete Grey-type characterization in the 2-cyclic two-type case, but the general necessary-and-sufficient criterion remains unresolved.