Decomposition and embedment of trajectories after explosion for a birth and death process (Q2503827)

This paper is a follow-up of three papers by the first named author on birth-and-death processes after explosion [Sci. China, Ser. A 41, No. 7, 694--699 (1998; Zbl 0915.60074); ibid. 42, No. 1, 38--47 (1999; Zbl 0928.46020); ibid. 43, No. 10, 1014--1018 (2000; Zbl 1054.60089)]. Consider the right-continuous birth-and-death process \(X(t)= X(t)(\omega)\), \(t<\sigma(\omega)\), with characteristic triple \((\alpha,C,D)\). Let \(U= U(\omega)\) be the set of all leaping intervals \([a,b)= [a,b)(\omega)\), i.e., \(a(\omega)\), \(b(\omega)\) are leaping points of \(X(t)(\omega)\) with no other leaping point in between, and define \(U_1:= U\setminus\{[0,\tau)\}\), \(\tau(\omega)\) being the first leaping point. For \(M\subset E:=\{0,1,2,\dots,\infty\}\) denote the set of all \([a,b)(w)\in U_1(\omega)\) such that \(X(a)(\omega)\in M\) by \({_MU_1}(\omega)\) and define \({_MU}:= {_MU_1}\cup\{[0,\tau)\}\), \({_MA}\) as the union of all intervals in \({_MU}\), and \({_MX}(t):= X(\inf\{u: \lambda({_MA}\cap [0, u))= t\})\), \(\lambda\) the Lebesgue measure. I.e., eliminate the sections of \(X\) corresponding to \({_{E\setminus M}U_1}\), keep those corresponding to \({_MU}\), translate them leftward according to their original order and connect them without intersecting them. The characteristic triple \(({_M\alpha}, {_MC}, {_MD})\) of \({_MX}\) is calculated, and \(X\) is decomposed into \(_{\{k\}}X\), \(k\in E\).