Difference between revisions of "Bellman-Harris process"
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[[Category:Branching processes]] | [[Category:Branching processes]] | ||
| + | A ''Bellmann Harris process'' is | ||
| + | a special case of an age-dependent branching process (cf. | ||
| + | [[Branching process, age-dependent|Branching process, age-dependent]]). It was first studied by R. Bellman and T.E. Harris {{Cite|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 | ||
| − | + | $$F(t,s) = \int_0^th(F(t-u,s))dG(u) + s(1-G(t)).$$ | |
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If | If | ||
| − | + | $$G(t)=1-e^{-\lambda t},\quad t\ge 0,$$ | |
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the Bellman–Harris process is a Markov branching process with continuous time. | the Bellman–Harris process is a Markov branching process with continuous time. | ||
====References==== | ====References==== | ||
| − | {| | + | {| |valign="top"|{{Ref|BH}}|| R. Bellman, T.E. Harris, "On the theory of age-dependent stochastic branching processes" ''Proc. Nat. Acad. Sci. USA'' , '''34''' (1948) pp. 601–604 {{MR|0027466}} |} |
| − | |valign="top"|{{Ref|BH}}|| R. Bellman, T.E. Harris, "On the theory of age-dependent stochastic branching processes" ''Proc. Nat. Acad. Sci. USA'' , '''34''' (1948) pp. 601–604 {{MR|0027466}} | ||
| − | |} | ||
Revision as of 19:31, 28 October 2014
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.
References
Bellman-Harris process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bellman-Harris_process&oldid=26348