Bellman-Harris process

2020 Mathematics Subject Classification: Primary: 60J80 [MSN][ZBL] A Bellmann Harris process is a special case of an age-dependent branching process (cf. Branching process, age-dependent). It was first studied by R. Bellman and T.E. Harris [BH]. In the Bellman–Harris process it is assumed that particles live, independently of each other, for random periods of time, and produce a random number of new particles at the end of their life time. If $G(t)$ is the distribution function of the life times of the individual particles, if $h(s)$ is the generating function of the number of direct descendants of one particle, and if at time $t=0$ the age of the particle was zero, then the generating function $F(t,s)={\rm E}s^{\mu(t)}$ of the number of particles $\mu(t)$ satisfies the non-linear integral equation

$$F(t,s) = \int_0^th(F(t-u,s))dG(u) + s(1-G(t)).$$ If

$$G(t)=1-e^{-\lambda t},\quad t\ge 0,$$ the Bellman–Harris process is a Markov branching process with continuous time.