Branching processes, regularity of

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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

B.A. Sewastjanow, "Verzweigungsprozesse" , Akad. Wissenschaft. DDR (1974) (Translated from Russian) MR0408018 Template:ZBL

Comments

Additional references can be found in the article Branching process.