Uniqueness and extinction properties of generalised Markov branching processes (Q1856867)

If in a continuous-time Galton-Watson process branching of a particle occurs with rate \(r> 0\), and the system is in the state \(i\), then the next branching event occurs with rate \(ir\). \textit{R.-R. Chen} [J. Appl. Probab. 34, No. 1, 14-23 (1997; Zbl 0874.60078)] discussed a generalization of this classical model by assuming that for a fixed number \(\nu> 0\) the next branching event occurs with rate \(i^\nu r\) instead, for all \(i\geq 0\). In the present paper it is shown that this leads to a unique process. Extinction properties are also studied.