Down/up crossing properties of weighted Markov collision processes (Q2048175)

The current paper focuses on weighted Markov collision processes, shortly WMC, which are continuous-time Markov chains with state space \(\mathbb{Z}_{+}\) and transition matrix defined in terms of births rates \((b_j)_{j\geq 0}\) and weights \((w_j)_{j\geq 0}\). For WMC, the authors investigate down/up crossing numbers. More precisely, in the main result Theorem 3.2 the joint probability generating function of down crossing and up crossing numbers for WMC is obtained. For making the understanding easier, several examples are considered towards the end of the work.