# 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 .

#### References

[1] | B.A. [B.A. Sevast'yanov] Sewastjanow, "Verzweigungsprozesse" , Akad. Wissenschaft. DDR (1974) (Translated from Russian) |

#### Comments

Additional references can be found in the article Branching process.

**How to Cite This Entry:**

Branching processes, regularity of.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Branching_processes,_regularity_of&oldid=11240