Branching processes, regularity of
A property of branching processes ensuring that the number of particles at any moment of time is finite. The problem of the regularity of a branching process is usually reduced to the problem of uniqueness of the solution of some differential or integral equation. For instance, in a continuous-time branching process the differential equation
with the initial condition has a unique solution if and only if, for any , the integral
is divergent. In the branching Bellman–Harris process the generating function of the number of particles is the solution of the non-linear integral equation
where is the distribution function of the lifetimes of particles and is the generating function of the number of daughter particles ( "direct descendants" ) of a single particle. If, for given and an integer , the inequalities
are valid for all , the solution of equation (*) is unique if and only if the equation
with initial conditions
has a unique solution
For a branching process described by equation (*) to be regular, it is necessary and sufficient for the integral
to diverge for any .
|[S]||B.A. Sewastjanow, "Verzweigungsprozesse" , Akad. Wissenschaft. DDR (1974) (Translated from Russian) MR0408018 Zbl 0291.60039|
Additional references can be found in the article Branching process.
Branching processes, regularity of. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Branching_processes,_regularity_of&oldid=26373