Bellman-Harris process

[ 2010 Mathematics Subject Classification MSN: 60J80 | MSCwiki: 60J80  ]

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 [1]. 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 is the distribution function of the life times of the individual particles, if is the generating function of the number of direct descendants of one particle, and if at time the age of the particle was zero, then the generating function of the number of particles satisfies the non-linear integral equation

If

the Bellman–Harris process is a Markov branching process with continuous time.

References

[1]	R. Bellman, T.E. Harris, "On the theory of age-dependent stochastic branching processes" Proc. Nat. Acad. Sci. USA , 34 (1948) pp. 601–604