Determine conservativity in the unresolved two-type branching-mechanism regime

Determine whether a two-type continuous-state branching process is conservative when its branching mechanism satisfies simultaneously: both diagonal Grey integrals diverge; at least one diagonal branching function is nonzero; and at least one of the corresponding diagonal integrals along the full diagonal diverges while the integral involving the sum of the two diagonal functions converges.

Background

The paper identifies the two-type cases not covered by its general sufficient conditions and its complete 2-cyclic characterization. The unresolved regime is described through three simultaneous conditions on the diagonal branching functions and their restrictions to the diagonal of the nonnegative quadrant.

If the first condition fails, a preceding corollary implies non-conservativity; if the second fails, the 2-cyclic Grey-type theorem applies; and if the relevant integrals in the third condition have the opposite behavior, the general sufficient conditions decide the issue. Thus, only the stated combination remains undecided by the results of the paper.