# Branching processes, regularity of

2020 Mathematics Subject Classification: Primary: 60J80 [MSN][ZBL]

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

$$\frac{\partial F (t; s) }{\partial t } = \ f (F (t; s))$$

with the initial condition $F(0; s) = s$ has a unique solution if and only if, for any $\epsilon > 0$, the integral

$$\int\limits _ {1 - \epsilon } ^ { 1 } { \frac{dx}{f (x) } }$$

is divergent. In the branching Bellman–Harris process the generating function $F(t; s)$ of the number of particles is the solution of the non-linear integral equation

$$\tag{* } F (t; s) = \ \int\limits _ { 0 } ^ { t } h (F (t - u; s)) dG (u) + s (1 - G (t)),$$

where $G(t)$ is the distribution function of the lifetimes of particles and $h(t)$ is the generating function of the number of daughter particles ( "direct descendants" ) of a single particle. If, for given $t _ {0} , c _ {1} , c _ {2} > 0$ and an integer $n \geq 1$, the inequalities

$$c _ {1} t ^ {n} \leq G (t) \leq c _ {2} t ^ {n}$$

are valid for all $0 \leq t \leq t _ {0}$, the solution of equation (*) is unique if and only if the equation

$$\frac{d ^ {n} \phi }{dt ^ {n} } = \ h ( \phi ) - 1$$

with initial conditions

$$\phi (0) = 1,\ \ \phi ^ {(r)} (0) = 0,\ r = 1 \dots n - 1,$$

has a unique solution

$$0 \leq \phi (t) \leq 1.$$

For a branching process described by equation (*) to be regular, it is necessary and sufficient for the integral

$$\int\limits _ { 0 } ^ \epsilon { \frac{dx}{x ^ {1-1/n } (1-h(1-x)) ^ {1/n} } }$$

to diverge for any $\epsilon > 0$.

#### References

 [S] B.A. Sewastjanow, "Verzweigungsprozesse" , Akad. Wissenschaft. DDR (1974) (Translated from Russian) MR0408018 Zbl 0291.60039